In an interference fit, also known as a press fit, two parts are joined together with minimal space in between. Initially, the inner object has a slightly larger diameter than the outer one. The outer part is heated until it is large enough to fit the inner part. After the inner part is inserted, the outer object shrinks over it during cooling, which strengthens the fastening. A suboptimal interference fit can cause the parts to come loose, bulge excessively because of permanent plastic deformations, or make it impossible for the parts to fit together. However, with the right interference fit, the parts essentially fuse into one piece that can be held together, even when met with a large amount of torque.
A new take on the popular Goldilocks fairytale: Instead of a bowl of porridge heated to her preferred temperature, Goldilocks finds an interference fit for two pipes that is neither too tight nor too loose.
An optimized interference fit reduces unwanted effects in a structure. For instance, in ball bearing assemblies, the interference fit protects against the bearing sliding on the shaft. If the interference fit is too loose, sliding will still occur, but if it is too tight, the ball bearing will experience increased operation temperature and wear particles.
For the best performance of a structure, the interference fit needs to be optimized with the application in mind. By building a simulation app, it is possible to efficiently evaluate parameters that affect the interference fit between two parts.
The Interference Fit Calculator computes and visualizes the interference fit between two pipes. The app includes an Input section where app users can enter different geometry parameters to quickly and easily test designs.
In this example, the inputs include:
The userfriendly interface of the app makes it easy to compute and visualize the results of the interference fit analysis. You can include buttons such as Solve, Reset to Default, Create Report, and Open Documentation for app users to run and view different analyses.
The different Results tabs of the app enable you to visualize how slight parameter changes affect the interference fit, which are computed by the underlying model. The results show the maximum transferable torque and axial force, as well as the effective stress, contact pressure, and pipe deformation for different inputs.
The Interference Fit Calculator is an example of what you can create with the Application Builder, a builtin tool included with the COMSOL Multiphysics® software. As you are in control of an app’s design, you can include different inputs and outputs to suit your needs.
The Interference Fit Calculator in action.
With a simulation app, you can test different parameters to optimize the interference fit for your specific structural application.
Click the button below to try the Interference Fit Calculator example.
One common treatment for atherosclerosis is a procedure called percutaneous transluminal angioplasty, which removes or compresses unwanted plaque that has built up in a patient’s coronary artery. This procedure sometimes relies on stents, placed within a blocked artery by an angioplasty balloon.
After reaching the intended location, the balloon inflates the stent, which locks into an expanded position. The balloon is then deflated and removed, while the stent remains in the artery. The expanded stent functions like a scaffold, keeping the blood vessel open and enabling blood to flow normally.
A stent example. Image by Lenore Edman — Own work. Licensed under CC BY 2.0, via Flickr Creative Commons.
Of course, for the angioplasty procedure to be a success, the tools used must perform as intended. If the ends of the stent expand more than its middle — a common defect known as dogboning — the artery can face serious damage. Another potential issue is foreshortening, which makes it challenging to position the stent and can also damage the artery.
To avoid these issues and make the angioplasty a success, it’s necessary to evaluate stent designs. One step in this process is analyzing the deformation experienced by a stent.
For this example, let’s examine a PalmazSchatz stent model, the geometry of which is seen below. This model looks at the stress and deformation in a stainless steel stent that is expanded via a radial outward pressure on the tube’s inner surface. (The pressure represents the balloon expansion.) The original diameter of the stent is 0.74 mm, but after the expansion period, the middle section has a diameter of 2 mm.
Thanks to the inherent symmetry of the stent’s geometry, we can minimize the computational costs of this simulation by reducing the size of the model to 1/24 of its original geometry.
The full stent geometry. The reduced geometry used in this example is represented by the darker meshed area.
First, let’s look at the various stresses and strains experienced by the stent during operation. Below, we see the stress distribution in the stent at maximum balloon inflation (left) and the residual stress in the stent after balloon deflation (right). As expected, stress in the stent is reduced after the balloon deflates.
Stress in the stent during balloon expansion (left) and after balloon deflation (right).
Moving on, we analyze how the effects of dogboning (blue) and foreshortening (green) change in relation to pressure during balloon inflation. Using this plot, we can check for potentially harmful effects in the stent design and optimize its performance.
Dogboning and foreshortening in the stent vs. the pressure in the angioplasty balloon.
We also examine the effective plastic strains in the tube at maximum dogboning, as seen in the following image.
Effective plastic strains and deformation at the time of maximum dogboning. The peak value is about 25%.
In regard to the recoil parameters, note that the longitudinal recoil is around −0.9%, the distal radial recoil is about 0.4%, and the central radial recoil is approximately 0.7%. These parameters provide more details on how the stent behaves when the inflated balloon is removed.
With the information provided by simulations like this one, engineers can improve the design of stents and optimize their use in biomedical applications. To try this example for yourself, click on the button below.
During the TPV cell energy production process, fuel burns within an emitting device that intensely radiates heat. Photovoltaic (PV) cells capture this radiation and convert it into electricity, with an efficiency of 1–20%. The required efficiency depends on the intended application of the cell. For example, efficiency is not a major factor when TPVs are used to cogenerate electricity within heat generators. On the other hand, efficiency is critical when TPVs are used as electric power sources for vehicles.
Left: Simplified schematic depicting the electricity generation process of a TPV. Right: An image from a prototype TPV system. Right image courtesy Dr. D. Wilhelm, Paul Sherrer Institute, Switzerland.
To improve the efficiency of TPV systems, engineers need to maximize radiative heat transfer, but this comes with a catch. The more radiation in the system, the less radiation converted to electric power. These losses — as well as conductive heat transfer — raise the temperature of the PV cell. If the temperature increases too much, it can exceed the operating temperature range of the PV cell, causing it to stop functioning.
One option for increasing the operation temperature of a TPV system is to use highefficiency semiconductor materials, which can withstand temperatures up to 1000°C. Since these materials tend to be expensive, engineers can reduce costs by combining smallerarea PV cells with mirrors that focus radiation onto the cells. Of course, there is a limit to how much the beams can be focused, since the cells overheat if the radiation intensity gets too high.
Engineers designing TPV devices need to find optimal system geometries and operating conditions that maximize performance, minimize material costs, and ensure that the device temperature stays within the operating range. Heat transfer simulation can help achieve these design goals.
This example uses the Heat Transfer Module and the SurfacetoSurface Radiation interface to determine how operating conditions (e.g., the flame temperature) affect the efficiency of a normal TPV system as well as the temperature of the system’s components. The goal is to maximize surfacetosurface radiative heat fluxes while minimizing conductive heat fluxes. In this model, the effects of geometry changes are also evaluated.
The model geometry includes an emitter, mirrors, insulation, and a PV cell that is cooled by water on its back side. For details on setting up this model — including how to add conduction, surfacetosurface radiation, and convective cooling — take a look at the TPV cell model documentation.
The TPV system model geometry.
To minimize the computational costs of the simulation, we use sector symmetry and reflection to reduce the computational domain to one sixteenth of the original geometry. When modeling the surfacetosurface radiation, we expand this view to account for the presence of all of the surfaces in the full geometry.
First, let’s check the voltaic efficiency of the PV cell for a range of cell temperatures. In doing so, we see that the efficiency decreases as the temperature increases. When the temperature of the cell exceeds 1600 K, the efficiency is 0. As such, the maximum operational temperature for the PV cell design is 1600 K.
Plotting PV cell voltaic efficiency versus temperature.
In the next plots, we see how the temperature of the emitter affects the temperature of the PV cell and the electric output power. The cell temperature plot (left image below) indicates that the emitter temperature must be under ~1800 K to keep the PV cell below its maximum operating temperature of 1600 K.
Keeping this in mind, let’s take a look at the electric power output results (right image below). From the results, we conclude that the maximum electric power is achieved when the emitter temperature is ~1600 K.
Plotting PV cell temperature (left) and electric output power (right) against operating temperature.
Moving on, let’s examine the temperature distribution in the PV cell for the optimal operating condition (left image below) and compare it to a temperature that exceeds this operating temperature (right image below). The two plots highlight how the device’s temperature distribution varies due to operating conditions.
The stationary temperature distribution in the full TPV system when the emitter temperature is 1600 K (left) and 2000 K (right).
Looking closer at the plot of the optimal emitter temperature of 1600 K, we see that the PV cells are heated to a sustainable temperature of slightly above 1200 K. It is important to note that the outside part of the insulation reaches a temperature of 800 K, indicating that a large amount of heat is transferred to the surrounding air. In addition, the irradiative flux significantly varies around the PV cell circumference and insulation jacket.
To determine the cause of this variation, we generate a plot of the irradiative flux for a single sector of symmetry at a temperature of 1600 K. The graph indicates that the variation is caused by shadowing and is related to the mirror positions. Using this plot, we could optimize the cell size and placement of the mirrors for a PV design.
The irradiation flux at the TPV cell, insulation inner surface, mirrors, and emitter.
Using models like the one discussed here, engineers can efficiently find optimal operating conditions for TPV devices, minimizing prototype development and testing.
To try this TPV cell example yourself, download the model files above.
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Picture a micromirror as a single string on a guitar. The string is so light and thin that when you pluck it, the surrounding air dampens the string’s motion, bringing it to a standstill.
Because this damping effect is important to many MEMS devices, micromirrors have a wide variety of potential applications. For instance, these mirrors can be used to control optic elements, an ability that makes them useful in the microscopy and fiber optics fields. Micromirrors are found in scanners, headsup displays, medical imagers, and more. Additionally, MEMS systems sometimes use integrated scanning micromirror systems for consumer and telecommunications applications.
Closeup view of an HDTV micromirror chip. Image by yellowcloud — Own work. Licensed under CC BY 2.0, via Flickr Creative Commons.
When developing a micromirror actuator system, engineers need to account for its dynamic vibrating behavior and damping, both of which greatly affect the operation of the device. Simulation provides a way to analyze these factors and accurately predict system performance in a timely and costefficient manner.
To perform an advanced MEMS analysis, you can combine features in the Structural Mechanics Module and Acoustics Module, two addon products to COMSOL Multiphysics. Let’s take a look at frequencydomain (timeharmonic) and transient analyses of a vibrating micromirror.
We model an idealized system that consists of a vibrating silicon micromirror — which is 0.5 by 0.5 mm with a thickness of 1 μm — surrounded by air. A key parameter in this model is the penetration depth; i.e., the thickness of the viscous and thermal boundary layers. In these layers, energy dissipates via viscous drag and thermal conduction. The thickness of the viscous and thermal layers is characterized by the following penetration depth scales:
where is the frequency, is the fluid density, is the dynamic viscosity, is the coefficient of thermal conduction, is the heat capacity at constant pressure, and is the nondimensional Prandtl number.
For air, when the system is excited at a frequency of 10 kHz (which is typical for this model), the viscous and thermal scales are 22 µm and 18 µm, respectively. These are comparable to the geometric scales, like the mirror thickness, meaning that thermal and viscous losses must be included. Moreover, in real systems, the mirrors may be located near surfaces or in close proximity to each other, creating narrow regions where the damping effects are accentuated.
The frequencydomain analysis provides insight into the frequency response of the system, including the location of the resonance frequencies, Qfactor of the resonance, and damping of the system.
The micromirror model geometry, showing the symmetry plane, fixed constraint, and torquing force components.
In this example, we use three separate interfaces:
By modeling the detailed thermoviscous acoustics and using the Thermoviscous Acoustics, Frequency Domain interface, we can explicitly include thermal and viscous damping while solving the full linearized NavierStokes, continuity, and energy equations. In doing so, we accomplish one of the main goals for this model: accurately calculating the damping experienced by the mirror.
To set up and combine the three interfaces, we use the AcousticsThermoviscous Acoustics Boundary and ThermoviscousAcousticsStructure Boundary multiphysics couplings. We then solve the model using a frequencydomain sweep and an eigenfrequency study. These analyses enable us to study the resonance frequency of the mirror under a torquing load in the frequency domain.
Let’s take a look at the displacement of the micromirror for a frequency of 10 kHz and when exposed to the torquing force. In this scenario, the displacement mainly occurs at the edges of the device. To view displacement in a different way, we also plot the response at the tip of the micromirror over a range of frequencies.
Micromirror displacement at 10 kHz for phase 0 (left) and the absolute value of the zcomponent of the displacement field at the micromirror tip (right).
Next, let’s view the acoustic temperature variations (left image below) and acoustic pressure distribution (right image below) in the micromirror for a frequency of 11 kHz. As we can see, the maximum and minimum temperature fluctuations occur opposite to one another and there is an antisymmetric pressure distribution. The temperature fluctuations are closely related to the pressure fluctuations through the equation of state. Note that the temperature fluctuations fall to zero at the surface of the mirror, where an isothermal condition is applied. The temperature gradient near the surface gives rise to the thermal losses.
Temperature fluctuation field within the thermoviscous acoustics domain (left) and the pressure isosurfaces (right).
The two animations below show a dynamic extension of the frequencydomain data using the timeharmonic nature of the solution. Both animations depict the mirror movement in a highly exaggerated manner, with the first one showing an instantaneous velocity magnitude in a cross section and the second showing the acoustic temperature fluctuations. These results indicate that there are highvelocity regions close to the edge of the micromirror. We determine the extent of this region into the air via the scale of the viscous boundary layer (viscous penetration depth). We can also identify the thermal boundary layer or penetration depth using the same method.
Animation of the timeharmonic variation in the local velocity.
Animation of the timeharmonic variation in the acoustic temperature fluctuations.
When the problem is formulated in the frequency domain, eigenmodes or eigenfrequencies can also be identified. From the eigenfrequency study (also performed in the model), we can determine the vibrating modes, shown in the animation below (only half the mirror is shown as symmetry applies). Our results show that the fundamental mode is around 10.5 kHz, with higher modes at 13.1 kHz and 39.5 kHz. The complex value of the eigenfrequency is related to the Qfactor of the resonance and thus the damping. (This relationship is discussed in detail in the Vibrating Micromirror model documentation.)
Animation of the first three vibrating modes of the micromirror.
As of version 5.3a of the COMSOL® software, a different take on this example solves for the transient behavior of the micromirror. Using the same geometry, we extend the frequencydomain analysis into a transient analysis. To achieve this, we swap the frequencydomain interfaces with their corresponding transient interfaces and adjust the settings of the transient solver. In the simulation, the micromirror is actuated for a short time and exhibits damped vibrations.
The resulting model includes some of the most advanced air and gas damping mechanisms that COMSOL Multiphysics has to offer. For instance, the Thermoviscous Acoustics, Transient interface generates the full details for the viscous and thermal damping of the micromirror from the surrounding air.
In addition, by coupling the transient perfectly matched layer capabilities of pressure acoustics to the thermoviscous acoustics domain, we can create efficient nonreflecting boundary conditions (NRBCs) for this model in the time domain.
Let’s start with the displacement results. The 3D results (left image below) visualize the displacement of the micromirror and the pressure distribution at a given time. We also generate a plot (right image below) to illustrate the damped vibrations caused by thermal and viscous losses. The green curve represents the undamped response of the micromirror when the surrounding air is not coupled to the mirror movement. The timedomain simulations make it possible to study transients of the system, like the decay time, and the response of the system to an anharmonic forcing.
Micromirror displacement and pressure distribution (left) and the transient evolution of the mirror displacement (right).
We can also examine the acoustic temperature variations surrounding the micromirror. The isothermal condition at the micromirror surface produces an acoustic thermal boundary layer. As with the frequencydomain example, the highest and lowest temperatures are located opposite to one another.
In addition, by calculating the acoustic velocity variations of the micromirror, we see that a noslip condition at the micromirror surface results in a viscous boundary layer.
Acoustic temperature variations (left) as well as acoustic velocity variations for the xcomponent (center) and zcomponent (right).
These examples demonstrate that we can analyze micromirrors using advanced modeling features available in the Acoustics Module in combination with the Structural Mechanics Module. For more details on modeling micromirrors, check out the tutorials below.
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The French scientist Barré de SaintVenant formulated his famous principle in 1855, but it was more of an observation than a strict mathematical statement:
“If the forces acting on a small portion of the surface of an elastic body are replaced by another statically equivalent system of forces acting on the same portion of the surface, this redistribution of loading produces substantial changes in the stresses locally, but has a negligible effect on the stresses at distances which are large in comparison with the linear dimensions of the surface on which the forces are changed.”
B. SaintVenant, Mém. savants étrangers, vol. 14, 1855.
Portrait of SaintVenant. Image in the public domain, via Wikimedia Commons.
Many great minds within the field of applied mechanics — Boussinesq, Love, von Mises, Toupin, and others — were involved in stating SaintVenant’s principle in a more exact form and providing mathematical proofs for it. As it turns out, this is quite difficult for more general cases, and research on the topic is still ongoing. (The argumentation has at times been quite vivid.)
Let’s start with something quite simple: a thin rectangular plate with a circular hole at some distance from the loaded edge, which is being pulled axially. If we are interested in the stress concentration at the hole, then how important is the actual load distribution?
Three different load types are applied at the rightmost boundary:
As seen in the plots below, the stress distribution at the hole is not affected by how the load is applied. The key here is, of course, that the hole is far enough from the load.
Von Mises stress contours for the three load cases.
Another way of visualizing this scenario is by using principal stress arrows. Such a plot emphasizes the stress field as a flux and gives a good feeling for the redistribution.
Principal stress plot for the three load cases. Note that there is a singularity when a point load is used.
By graphing the stress along a line, we can see that all three cases converge to each other at a distance from the edge, which is approximately equal to the width of the plate.
Stress along the upper edge as a function of the distance from the loaded boundary. The distance is normalized by the width of the plate.
If the hole is moved closer to the loaded boundary, we get another situation. The stress state around the hole now depends on the load distribution. But even more interesting is that the distance to where the three stress fields agree now is twice as far from the loaded boundary. The application of SaintVenant’s principle requires that the stresses are free to redistribute. In this case, that redistribution is partially blocked by the hole.
Stress along the upper edge with the hole closer to the loaded boundary.
Note that SaintVenant’s principle tells us that there is no difference in the stress state at a distance that is of the order of the linear dimension of the loaded area. The loaded area to be taken into consideration, however, may not be the area that is actually loaded! This statement may sound strange, but think of it this way: When the hole is far away, we may compute the stress concentration factor using a handbook (mine says 4.32) rather than by an FE solution. The handbook approach contains an implicit assumption that the load is evenly distributed as in the first load case. So even if the actual load was applied to only a small part of the boundary, the critical distance in that case is related to the size of the whole boundary.
When solving the problem using the finite element method (FEM), then the hole can be arbitrarily close to the load. What sets the limit is that from the physical point of view, the load distribution is well defined. As soon as we make assumptions about redistribution, however, there is an implicit assumption about the load distribution, which may differ from the actual one.
So far, we have said that the stresses are the same independent of the load details at some suitable distance. Since we are dealing with linear elasticity here, it is always possible to superimpose load cases. When working with proofs of SaintVenant’s principle, it is easier to formulate a principle along these lines: The stresses caused by a load system with no resulting force or moment will be small at a distance that is of the same order of magnitude as the size of the loaded boundary.
Thus, we study the stress caused by the difference between the two load systems with equal resultants. Most modern proofs are based on estimates of the decay of the strain energy density for such a zeroresultant system.
Returning to the problem above, we can compute the difference between the load cases. Doing so allows us to study the actual decay of stress or strain energy density for the difference of the stress fields.
Logarithm of strain energy density for the zeroresultant load cases.
The strain energy density along the plate for the zero resultant load cases. The energy is integrated along the vertical direction in order to produce a quantity that is only a function of the distance from the load.
The decay in the logarithm of the strain energy density is more or less linear with the distance from the loaded boundary. This is actually in line with what modern proofs predict: an exponential decay of the strain energy density. We can also clearly see how the hole temporarily reduces the decay rate.
For thinner structures like shells, beams, and trusses, it is well known that SaintVenant’s principle cannot be applied the same way as for a more “solid” object. Disturbances travel longer distances than what we expect, because the load paths in a thin structure are much more limited. This is the same phenomenon we see with the hole in the example above, but more prominently.
Here, we study a beam with a standard IPE100 cross section. The end of the beam is subjected to an axial stress, with an amplitude that has a linear distribution in both crosssectional directions.
Load distribution, displayed as contours and arrows.
Due to the symmetries, this load has a zeroresultant force, as well as zero moment around all axes. The height of the cross section is 100 mm, so if the standard form of SaintVenant’s principle is applicable, then the stresses should be small at a distance of approximately 100 mm from the end section.
Effective stress in the beam. The red contour indicates where the stress is less than 5% of the peak applied stress.
It turns out that in order for the stress to be below 5% of the peak applied stress, we have to travel almost a meter along the beam. Thus, the load redistribution is much less efficient here, since the equilibration between the top and bottom flanges requires moment transfer through the thin web.
If you are familiar with the theory for nonuniform torsion of beams (i.e., warping theory or Vlasov theory), you will recognize that the applied load has a significant bimoment. The bimoment is a crosssectional quantity with the physical dimension force X length^{2}.
Maybe (this is just my personal speculation), an efficient SaintVenant’s principle for this case should require not only force and moment but also a bimoment of zero. This can be accomplished by adding four point loads that provide a counteracting bimoment. The result of such an analysis is shown below.
Effective stress with four point loads that also provide a zero bimoment. The 5% stress contour is now much closer to the loaded boundary.
The applied point loads, which are not optimally placed on purpose, give extremely high (actually singular) local stresses. However, the stress does drop off much faster and is below 5% after about 100 mm. The 5% limit is still in terms of the applied distributed load, so it is not adjusted for the new local stresses. The logarithmic decay rate of the strain energy density is three times faster after the point loads are added.
In some cases, you can intuitively consider SaintVenant’s principle to be applicable to the FE discretized problem. Here, we look at distributed loads and nonconforming meshes.
In the FE model, loads are always applied at the mesh nodes, even though you specify them as a continuous boundary load. The load is internally distributed to the nodes of the element using the principle of virtual work, as shown in the example below.
A linearly distributed load and how it is applied at the nodes of a secondorder Lagrange element with side length L.
There is, however, an infinite number of load distributions that give the same nodal loads as long as they share the same resultant force and moment. Obviously, the solution to the finite element problem is the same for all of these cases. From SaintVenant’s principle, however, we can conclude that all such loads should give essentially the same stress field as soon as we are some distance away.
Since the size of the area over which we redistribute loads is an element face, the linear dimension after which there is no difference is essentially one element layer inside the structure. Thus, the solution in the outermost layer of elements may not correspond to the actual load, but further in, it does.
As an example, we can load a rectangular plate with a boundary load that has an exponential stress distribution. The stress computed with a fine mesh is shown below.
Contour plot of the axial stress distribution.
Because of SaintVenant’s principle, the stress field is redistributed to a pure bending state at some distance from the loaded edge, just as we expect. This, however, is not the target of the current discussion. Rather, we investigate the difference between the stress distribution above, and what we get with a number of coarse meshes.
Error in axial stress for three different meshes. Note the different scales. As expected, the error is smaller when the mesh is finer.
As can be seen in the figure, the error quickly decreases after the first element layer. What we see here is actually a combination of mesh convergence and the redistribution of stresses implied by SaintVenant’s principle.
A nonconforming mesh occurs when the shape functions in two connected elements do not match. The most common case is when an assembly is connected using identity pairs and continuity conditions. To exemplify this, we can study a straight bar with an intentionally nonmatching mesh. With a simple load case, such as uniaxial tension, it is possible to study the stress disturbances caused by the transition.
Axial stress at a nonconforming mesh transition. Secondorder elements are used.
The forces transmitted by the nodes at the two sides do not match the assumption of constant stress. Again, this can be seen as a local load redistribution over an area that is the element size. Using the reasoning of SaintVenant, the disturbance should fade away at an “elementsized” distance from the transition. Let’s investigate what happens if the mesh is refined in the axial direction.
Region with more than 0.1% error in stress. Three different discretizations are used in the axial direction.
It turns out that the region of disturbance is not affected much by the discretization in the direction perpendicular to the transition boundary. This is exactly what SaintVenant’s principle tells us.
Without making use of SaintVenant’s principle, many structural analyses are difficult to perform, simply because the detailed load distribution is not known.
The principle is formally only valid for linear elastic materials. In practice, we also intuitively use it on a daily basis for other situations. If, for example, the material in the “plate with a hole” example were elastoplastic, we would expect the two distributed loads to give equivalent results, as long as the yield stress is above the stress applied at the boundary so that there is only plastic deformation around the hole. The point load, however, always gives a different solution, since the material yields around the loaded point. For a longer discussion, read this blog post on singularities at point loads.
Learn more about using the COMSOL Multiphysics® software for FEA.
Born in 1707 in Basel, Switzerland, Leonhard Euler (pronounced “oiler”) was a prolific mathematician who published more than 800 articles during his lifetime. He studied under the famous Johann Bernoulli and received his master’s degree in philosophy from the University of Basel. Before moving to St. Petersburg, Russia, to work at the university, Euler submitted his first paper to the Paris Academy of Sciences, coming in second place at only 19 years old.
A portrait of Leonhard Euler. Image in the public domain, via Wikimedia Commons.
Euler quickly rose through the academic ranks and in 1733 succeeded Bernoulli as the chair of mathematics in St. Petersburg. Euler moved to Berlin in 1741 at the invitation of King Frederick II. In his 25 years there, he wrote around 380 articles and the first volume of his seminal book Introductio in Analysin Infinitorum, which formally defined functions for the first time; introduced the notation; popularized the and notation; and established the critical formula .
JosephLouis Lagrange (pronounced “luhgronj”) was born Giuseppe Lodovico Lagrangia in Turin. Today, this city is the capital of the region of Piedmont in Italy, but when Lagrange was born in 1736, it was ruled by the Duke of Savoy as part of the Kingdom of Sardinia. Lagrange developed an interest in mathematics and, after working independently on novel topics, began corresponding with Euler, whom he succeeded when Euler left Berlin.
A portrait of JosephLouis Lagrange. Image in the public domain, via Wikimedia Commons.
In Berlin, Lagrange developed most of the mathematics for which he is famous today. He played an important role in the development of variational calculus and came up with the Lagrangian approach to mechanics. Although Lagrangian mechanics makes the same predictions as Newton’s laws of motion, the Lagrangian functional introduced by Lagrange allows the classical mechanics of many problems to be described in a mathematically more straightforward and insightful manner than in Newtonian mechanics. Lagrange also developed the method of Lagrange multipliers, which allows constraints on systems of equations to be introduced easily in a variational approach.
The mathematical formulations of Euler and Lagrange are fundamental to the finite element method, which is used to solve equations in COMSOL Multiphysics.
In the Eulerian method, the dynamics of a system are considered from the viewpoint of an observer measuring the system’s evolution with respect to a fixed system of coordinates. This coordinate system is called the spatial frame in COMSOL Multiphysics. It could be understood to correspond to the laboratory frame in physical analysis, since the system of coordinates is oriented according to a fixed set of axes without any reference to the orientation of the components of the physical system itself.
The figure below illustrates a thin plate of material whose structural mechanics are modeled in a 2D plane. The plate is fixed to a rigid wall at the lefthand side and is deformed under its own weight, as gravity acts downward. With the results plotted in the spatial frame, we see the deformation of the object, as we would expect to observe in the laboratory.
A thin plate fixed to the gray block at the left deforms under its own weight, as viewed in the spatial (lab) frame. The deflection at the tip is about 5 mm for the given mechanical properties.
Formulating physical equations seems very natural in the Eulerian method. Indeed, this is the common formulation for problems such as electromagnetics and fluid physics, in which the field variables are expressed as functions of the fixed coordinates in the spatial frame.
For mechanical problems, though, the Lagrangian method offers a helpful alternative. In the Lagrangian method, the mechanical equations are written with reference to small individual volumes of the material, which will move within an object as it displaces or deforms dynamically. To put it another way, the object itself always appears undeformed from the point of view of the Lagrangian coordinate system, since the latter stays attached to the deforming object and moves with it, but external forces in the surroundings appear to change their orientation from the deforming object’s perspective. The corresponding coordinate system, which moves along with the deforming object, is called the material frame in COMSOL Multiphysics.
A point within the object, as measured in the spatial frame, is displaced from the position of the same point as expressed in the material frame by the mechanical displacement of that point. In the image below, we focus our view on the tip of the deforming plate in the example above and animate its deformation as the density of the object increases so that the weight increases too. As you can see, the material frame coordinate system (red grid and arrows) deforms together with the object, as the object’s dimensions in the spatial frame change. This means that anisotropic material properties — such as mechanical properties of composite materials — can be expressed conveniently in the material frame.
Zoomedin view of the tip of a thin plate deforming under its own weight, as its density is increased. The red grid denotes the material frame coordinates, tied to the object, as viewed in the spatial (lab) frame. The red and green arrows show the x and ycoordinate orientations of the material frame, as viewed in the spatial frame.
In the limit of very small strains for this type of mechanical problem, the spatial and material frames are nearly coincident, because the mechanical displacement is small compared to the object’s size. In this case, it is common to use the “engineering strain” to define the elastic stressstrain relation for the object, and the resulting stressstrain equations are linear. As the mechanical displacement increases, though, the linear approximation used to evaluate the engineering strain is increasingly inaccurate — the exact GreenLagrange strain is required. In COMSOL Multiphysics, the term “geometric nonlinearity” means that the GreenLagrange strain is used.
For further details on the mathematics, see my colleague Henrik Sönnerlind’s blog post on geometric nonlinearity.
Geometric nonlinearity is handled in COMSOL Multiphysics by allowing the spatial frame to be separated from the material frame, according to a frame transformation due to the computed mechanical displacement. It remains convenient to access the material frame to express properties such as anisotropic mechanical material properties, since these properties will usually remain aligned with the material frame coordinates, even as the object deforms.
By contrast, external forces such as gravity have a fixed orientation in the spatial frame. From the perspective of the material frame, external forces like gravity change direction as the object deforms. The image below shows the tip of the thin plate as above, but here, the displacement magnitude is plotted with colors. Arrows are used to illustrate the force due to gravity, as expressed in the material frame coordinates. Since the material frame coordinates remain fixed with respect to the object, the dimensions of the object appear not to change. However, the displacement magnitude increases with the object’s weight and the gravity force increasingly changes direction with respect to the deformed material in conditions of greater deformation.
Zoomedin view of the tip of a thin plate deforming under its own weight as its density increases. The plot is in the material frame as used for the Lagrangian formulation, so the deformation is not apparent, although displacement increases. The red arrows indicate the apparent direction of gravity (which is constant in the spatial frame) as perceived from the material frame of reference within the deforming object.
Neither the Lagrangian nor Eulerian formulation is more “physical” or “correct” than the other. They are simply different mathematical approaches to describing the same phenomena and equations. Through coordinate transformation, we can always transform the physical equations for any phenomenon from the material frame to the spatial frame or vice versa. From the perspective of interpretation and implementation, though, each approach has certain advantages and common applications. Some of these are summarized in the table below:
Strengths  Common Applications  

Eulerian Method 


Lagrangian Method 


What about multiphysics problems, such as fluidstructure interaction (FSI) or geometrically nonlinear electromechanics? In these cases, one physical equation might be formulated most naturally with the Eulerian method, while another might be better expressed with the Lagrangian method. This is where the ALE method comes in. This method solves the equations on a third coordinate system, which is not required to match either the spatial frame or the material frame coordinate systems.
The third coordinate system is called the mesh frame in COMSOL Multiphysics. There is one mathematical mapping between the spatial frame and the underlying mesh frame, and one between the material frame and the underlying mesh frame, so at all points in time, the equations formulated in the spatial and material frames can be transformed into the mesh frame to be solved.
In domains representing solids in a model, mechanical displacement is predicted using structural mechanics equations in the Lagrangian formulation. Here, the relation of the spatial and material frames is given by the mechanical displacement, as above. The ALE method adds more equations to allow the apparent positions and shapes of mesh elements in neighboring domains to displace in the spatial frame. That is in order to account for how mechanical deformation can change the shape of the boundaries of any domain where the physics are described in the Eulerian formulation. These additional equations are called a Moving Mesh or Deformed Geometry in COMSOL Multiphysics.
At boundaries between Lagrangian and Eulerian domains, a boundary condition for these additional equations requires that the displacement of the spatial frame (as defined through the moving mesh) for the Eulerian domain must match the mechanical displacement of the spatial frame away from the material frame in the Lagrangian domain. Even where no mechanical equations are solved, such that no Lagrangian method is used, the ALE method can still be used to express moving boundaries due to deposition or loss of material.
If you find the ALE method quite mathematical, that’s OK! It’s a difficult concept to follow in the abstract. To better understand the way the ALE method works, let’s take a look at an example within COMSOL Multiphysics.
The ALE method plays an important role in modeling FSI. In COMSOL Multiphysics, this method enables the automated bidirectional coupling of fluid flow and structural deformation, a capability demonstrated in our Micropump Mechanism tutorial model.
At the heart of this micropump mechanism are two cantilevers, which perform the same function as valves in conventional pumping devices. These cantilevers are flexible enough that the fluid flow causes them to deform. As fluid is alternately pumped into or out of the channel at the top, the force of the fluid flow causes the two cantilevers to deform so that fluid flows out to the right or in from the left.
The micropump mechanism. Pumping fluid into or out of the top tube produces opposite reactions in the two cantilevers, pushing fluid in or out of the chamber. Even though there is no timeaveraged net flow into the upper tube, there is a timeaveraged net movement of fluid from left to right.
The cantilevers deform enough that there is an appreciable change in the position of the boundary where the fluid and solid meet: a geometrically nonlinear case. The selfconsistent handling of the fluid’s pressure on the solid and the solid’s force on the fluid, together with the deformation of the mesh, are handled automatically by the FluidStructure Interaction interface. The interface employs the ALE method to account for the change in shape in the solid and fluid regions.
For solids, the mechanical equations with geometric nonlinearity define the displacement of the spatial frame with respect to the material frame. In the fluid equations, it’s necessary to deform the mesh on which the equations are solved in order to express the displacement of the solid boundaries in the spatial frame where the fluid equations are formulated. The deformation at the boundaries is controlled by the mechanical displacement from the solution to the structural problem. Within the fluid, though, the exact position or orientation of mesh nodes isn’t important, as the equations are formulated in the fixed spatial frame. Instead, the deformation of the mesh is smoothed in order to ensure that the numerical problem remains stable with highquality mesh elements.
To explain the ALE method for the FSI problem, we could paraphrase a common explanation for general relativity: forces due to fluid flow (Eulerian) tell the structure how to deform in the material frame (Lagrangian), while the structural deformation (Lagrangian) tells the mesh how to move in the spatial frame (Eulerian).
Top: The micropump’s operation, including pressure, flow, and cantilever deformation, as plotted in the spatial frame. Bottom: Mesh deformations calculated by the ALE method.
As of COMSOL Multiphysics version 5.3a, the Moving Mesh feature to define mesh deformation in this type of problem is located under Component > Definitions. This allows consistency in the definition of material and spatial frames between all physics included in a model, even if several physics interfaces are included. The screen capture below shows where these settings are located in the COMSOL Multiphysics Model Builder tree.
Screen capture showing Moving Mesh features under Component > Definitions, and physical coupling between two physics interfaces through Multiphysics > FluidStructure Interaction.
Turning to an electrochemical problem, the Copper Deposition in a Trench tutorial model shows that the ALE method can be vital for simulating electrodeposition problems. In this model, copper is deposited onto a circuit board that has a small “trench”. The deposited copper layer becomes thick compared to the overall size of the trench, so the size and orientation of the copper surface change appreciably as deposition proceeds. Since the rate of copper deposition at different points on this surface is nonuniform, the shape and movement of the boundary cannot be neglected.
A schematic of the physical problem being solved in the electrodeposition model.
To calculate the rate of deposition at a given point on the copper electrodeelectrolyte interface, we need the concentration of the species and the electrolyte potential of the solution adjacent to that point. As the deposition progresses and the boundary moves, the shape of the electrolyte volume has to change continuously. Similarly, the concentration and potential distributions on the altered shape must be recalculated.
The coupling of the deposition rate to the boundary motion rate and the calculation of the changing shape are accomplished with the ALE method and fully automated multiphysics couplings with the Tertiary Current Distribution and Deformed Geometry interfaces. Here, the Deformed Geometry displaces the copper surface in the spatial frame at a rate proportional to the local current density for electrodeposition, as computed from the electrochemical interface.
With this model, we can accurately account for the deposition process in order to optimize its parameters. We can also experiment with different applied potentials and deposition surface geometries to improve the uniformity of the deposition, which produces a more efficient process and a higherquality end product.
Animations showing the evolution of the deposition process in time. It is clear that the deposition happens unevenly, resulting in a pinching of the trench opening at its top.
Thermal ablation, discussed in this previous blog post, involves a very high temperature applied to an object, causing the surface to melt and vaporize. Examples of thermal ablation include the removal of material by lasers — such as in the etching process, laser drilling, or laser eye surgery — and a spacecraft’s heat shield as it reenters the atmosphere.
Animation showing the effect of thermal ablation on a material.
Since we expect that an object’s shape will change when some of its material is removed, deforming meshes are clearly a key part of thermal ablation simulation. What we need to know is how the shape of the object will change. This depends on how we balance the applied heat with heat lost to ablation and heat dissipation throughout the structure by mechanisms such as conduction.
To obtain this information, we can predict the temperature profile as a function of space and time by solving the heat transfer equations using the Heat Transfer interface. Because the mass and shape of the object are changing, the Heat Transfer interface is coupled to a Deformed Geometry interface, using the ALE method to displace the boundary according to the rate of ablation. The Heat Transfer equations predict the temperature distribution in the object as its shape evolves.
By performing these steps, we can attain accurate calculations for the thermal ablation process. Moreover, we can determine the final shape of the object after ablation is complete. This might enable us to check whether a laser weld will fall within acceptable tolerances or whether a spacecraft will survive an emergency landing.
The contributions of Leonhard Euler and JosephLouis Lagrange in the field of mathematics have paved the way for simulating a variety of systems involving multiphysics applications. The combination of their individual methods has led to the development of the ALE method, which can be used to predict physical behavior when objects deform or displace. By properly accounting for these movements, you can set up highly accurate models. Remember to thank Euler and Lagrange as you investigate these and other models that exploit the ALE method!
The ALE method is one of many builtin physics capabilities in the COMSOL Multiphysics® software. See more of them:
To measure blood flow in a painless and noninvasive way for patients, medical professionals can use magnetic flow meters, which rely on electromotive force (EMF). In a flow meter, external coils generate a magnetic field and noncontact electrodes measure the induced EMF. When a patient moves during the measurement process (even by merely taking a breath), blood vessels can move, which affects the flow meter’s sensitivity. This phenomenon is an important point of analysis in cardiac and thoracic medicine.
Magnetic flow meters, which rely on coils and electrodes, are a noninvasive way to measure flow in a patient’s blood vessels.
Researchers from ABB Corporate Research in India built a multiphysics model of this process that includes the effects of fluidstructure interaction (FSI) and electromagnetics. Their aim was to understand how blood vessel movement influences flow meter sensitivity by comparing the meter’s performance when blood vessels are displaced versus in their normal positions.
The research team modeled a blood vessel as a pipe, but with the appropriate biological material properties. They coupled multiple physical effects via builtin physics interfaces in the COMSOL Multiphysics® software. The Laminar Flow interface was used to model the blood flow through the vessel. To account for the magnetic field generated by the coils, as well as the EMF induced by the blood flow and magnetic field, they used the Magnetic and Electric Fields interface.
A schematic of the magnetic flow meter model. Image by S. Dasgupta, K. Ravikumar, P. Nenninger, and F. Gotthardt and taken from their COMSOL Conference 2016 Bangalore paper.
The researchers also used the Structural Mechanics Module, an addon product to COMSOL Multiphysics, to model the vessel displacement during patient movement. They coupled this analysis with the CFD Module to account for FSI, including how the vessel displacement affects blood flow and how the fluid pressure affects the blood vessel.
This model was used to analyze the sensitivity of the magnetic flow meter when blood vessels are displaced and when they are in a normal position.
The researchers compared the results for a blood vessel in a normal position and displaced by 5 cm toward the upper coil. The simulation results show contour plots of the velocity and magnetic flux density across the pipe (i.e., blood vessel) cross section. Other results show the induced electric potential from the flow and magnetic field interface and the potential distribution across the pipe diameter.
Induced electric potential across the pipe cross section (top left) and length (bottom left) as well as the velocity (top right) and magnetic flux density (bottom right) across the pipe cross section. Images by S. Dasgupta et al. and taken from their COMSOL Conference 2016 Bangalore paper.
The plots indicate an increase in the sensitivity of the flow meter between the nondisplaced and displaced blood vessels. The team determined that the increase is not due to velocity, since the velocity profiles are the same for both scenarios. Instead, the reason for the increase is that the displaced vessel shifted to a higher magnetic field zone, and the magnetic flux density in the vessel increased as it moved toward the coil.
The researchers from ABB concluded through their simulation results that blood vessel displacement is a potential issue for medical uses of magnetic flow meters. To address the concerns, they theorized that a patient’s body movement can be restricted during these procedures or a breathsynchronous magnetic field can be generated to compensate for the sensitivity changes.
While these simulations proved useful for the medical field, further studies can account for more realworld conditions, such as pulsatile blood flow as well as variations in vessel properties and body locations.
For the team, this research confirmed that COMSOL Multiphysics can be used to analyze how different phenomena — including fluid flow, structural mechanics, and electromagnetics — interact in a complex application.
Check out the full paper from the COMSOL Conference 2016 Bangalore (it won a Best Paper award!): “Measurement of Blood Flowrate in Large Blood Vessels Using Magnetic Flowmeter“
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Canadian Nuclear Laboratories aims to improve nuclear fuel because it limits the efficiency of power generation in nuclear reactors. As Andrew Prudil said in his keynote talk: “If we can increase the power rating of the reactors, that’s worth millions of dollars per day.” Optimized nuclear fuel also enables more green energy on a power grid and reduces the risk of nuclear accidents. Plus, the improved fuel can be used in existing reactors to enhance their performance.
Before engineers can develop improved nuclear fuel, they have to understand its behavior. This is no simple matter, as nuclear fuel experiences multiple physical phenomena during nuclear reaction fission, including high temperatures, radiation, mechanical loading, thermal expansion, the creation of fission products such as xenon and krypton, and more.
To learn more about nuclear fuel behavior during a reaction, in which “everything depends on everything else,” CNL turned to the COMSOL Multiphysics® software.
First, Prudil discussed a multiphysics model — created for his PhD thesis — that studies the behavior of nuclear fuel (or pellets, in this case). The Fuel and Sheath Modeling Tool (FAST) simulates a long row of pellets separated by small gaps inside a metal sheath. Each part of the model involves multiple types of physics. For instance, sheaths in nuclear reactors typically use zirconiumbased alloys, which consist of anisotropic crystal structures. For accurate results, the model must account for how the crystals behave when pulled in different directions.
From the video: Results for the FAST simulations.
The simulations show how the ends of the pellets push outward to make room for the hot material at the center. The “hourglassing” phenomenon causes the ends of the pellets to create a wavy pattern in the cladding (exaggerated in the image above). FAST can also plot the radial displacement and various stress and strain fields, such as the hydrostatic pressure, von Mises stress, and axial creep. Prudil noted that the results show “very interesting, very rich spatial fields.”
With FAST, it’s possible to look at how nuclear fuel behaves in a continuum — both in terms of a temperature gradient and mechanical loading.
Prudil then discussed a model created at Canadian Nuclear Laboratories that simulates how fission gas forms bubbles on the boundary of a single grain of uranium oxide, a process that involves fission gas products such as xenon and krypton. At the grain boundary, these insoluble gases try to diffuse pressure by forming bubbles. The bubbles grow larger and larger and let gases escape.
The CNL model simulates this process for individual bubbles. Instead of using the traditional phase field method, which can be computationally expensive, they created the included phase technique to model the phase interface.
Simulation results for the included phase technique. Animation courtesy Andrew Prudil and can be found in the paper: “A novel model of third phase inclusions on two phase boundaries“.
Initially, the simulations show a random distribution of bubbles on the grain boundary. As time progresses, the bubbles combine to minimize the surface energy before collecting at the edges and vertices. CNL validated their approach, determining that they could control the contact angle of a single bubble on an infinite plane.
Wrapping up, Prudil mentioned that COMSOL Multiphysics could also be used to investigate other interesting multiphysics phenomena (e.g., columnar grain growth). With these capabilities, engineers can learn more about nuclear fuel and continue to advance the field.
To learn more about how CNL uses multiphysics modeling to understand the behavior of nuclear fuel, watch the keynote video at the top of this post.
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The Nonlinear Structural Materials Module, an addon product to COMSOL Multiphysics®, provides a plethora of material models, including models for hyperelasticity, isotropic/kinematic hardening plasticity, viscoelasticity, creep, porous plasticity, soils, and more. These material models cover a vast majority of engineering problems within structural analysis.
However, in some situations, the mechanical behavior of a material is not readily expressed in terms of an existing material model. For instance, suppose you developed a specialized material model for a certain alloy and want to use it to solve a large structural mechanics boundary value problem in COMSOL Multiphysics. What do you do?
As a matter of fact, there are three different ways in which you can define your own material:
The implementation of a material model as an external DLL can seem like a complex endeavor, but this blog post demonstrates how to implement an elastoplastic material model in COMSOL Multiphysics using handson steps that you can follow.
As a starting point, we need to decide on a material model to implement. We choose an isotropic linearelastic material with isotropic hardening. This is a simple plasticity model that already exists in COMSOL Multiphysics, but it serves nicely to convey some key points.
First, let’s go over some assumptions, definitions, and nomenclature:
The example material model: Uniaxial stressstrain curve and yield surface in principal stress space.
Now, let’s discuss approaches for implementing a material model as an external material. There are several different ways of calling usercoded routines for external materials, which we refer to as sockets.
We can use the General stressstrain relation socket of the external material to define a complete material model that includes (possibly) both elastic and inelastic strain contributions. This is the most general of the two modeling approaches discussed here. When we use the General stressstrain relation socket, we are faced with two tasks:
We can also use the Inelastic residual strain socket to define a description of an inelastic strain contribution to the overall material model. An example of this would be if we wanted to add our own creep strain term to the builtin linear elastic material. The Inelastic residual strain socket assumes an additive decomposition of the total (GreenLagrange) strain into elastic and inelastic parts. Thus, this is an adequate assumption when strains are of the order < 10%. When we use the Inelastic residual strain socket of the external material model, we are faced with two tasks:
Two related External Material sockets are the General stressdeformation relation and the Inelastic residual deformation. These are more general versions of those discussed above. Instead of defining the deformation in terms of the GreenLagrange strain tensor, the deformation gradient is provided. Many largestrain elastoplastic material models use a multiplicative decomposition of the deformation gradient into elastic and plastic parts. In these situations, you would likely want to use one of these sockets instead.
Tip: We link to the source file and model file at the bottom of this blog post.
The complexity of computing the stress tensor varies significantly between material models. In practice, the computation of the stress tensor often needs to be formulated as an algorithm. This is often called a stress update algorithm in literature. In essence, the objective of a stress update algorithm for a material model is to compute the stresses, knowing:
These quantities are provided to the external material as input.
The term “material state” represents any solutiondependent internal variables that are required to describe the material. Examples of such variables are plastic strain tensor components, current yield stress, back stress tensor components, damage parameters, effective plastic strain, etc. The choice of such state variables will depend on the material model. We must ensure that the material state is properly initialized at the start of the analysis, and that it is updated at the end of the increment.
We first need to investigate if there is plastic flow occurring during the increment. We do this by assuming that the elastic strain is equal to the total strain of the current increment, less the (deviatoric) plastic strain of the previously converged increment, . This assumption would hold true if there was, indeed, no plastic flow during the increment. The deviatoric stress tensor that is computed this way is aptly called a trial stress deviator and is given by
with an effective (von Mises) value .
The effective trial stress is compared to the yield stress of the material by assuming that there is no plastic flow during the increment. The yield stress corresponding to the previously converged increment is given by
Notice that the leftsuperscripted and in Steps 1 and 2 represent the material state of the previously converged increment, as we discussed earlier.
Now we check if the trial stress causes plastic flow. For another way of expressing this, if the trial stress is inside the yield surface, the response will be purely elastic during the increment. If not, plastic flow will result. The check is performed using the yield condition:
Check of the yield condition to determine elastic or elastoplastic computation.
The stress update algorithm now necessarily branches off into either a purely elastic computation or an elastoplastic computation. We will follow each of these branches, starting with the purely elastic branch.
Because we determined that there is no plastic flow during the increment, the trial stress deviator is, in fact, identical to the stress deviator, and the update of the plastic strain tensor and the effective plastic strain is trivial.
We can directly return the pure elastic stressstrain relation as the Jacobian.
The objective of the elastoplastic branch of the stress update algorithm is to compute the stress deviator and update the plastic strains. We begin by again expressing the stress deviator, now knowing that plastic flow takes place during the increment:
or
In the above equation, we used a discrete form for the flow rule that states that an increment in plastic strain is proportional to the stress deviator through a socalled plastic multiplier . Let’s stop for a moment and consider a graphical representation of this equation for stress:
Graphical representation of the correction of the trial stress deviator.
If we compute a trial stress deviator that lies outside the yield surface, we need to make a correction so that the stress deviator is returned to the tobedetermined yield surface. The plastic multiplier determines the exact amount by which the trial stress deviator should be scaled back to give the correct stress deviator. If we compute the plastic multiplier, it is straightforward to then compute the stress deviator and the plastic strain increment.
The key steps are to:
We can relate the plastic multiplier to the effective plastic strain increment using the flow rule and then transform the equation for stress into a governing scalar equation:
This is in general a nonlinear equation, and we need to solve it using a suitable iterative scheme. We are now ready to compute the stress tensor, the plastic strain tensor, and the effective plastic strain.
The updated plastic strain tensor and effective plastic strain are stored as state variables.
We computed the stresses and updated the material state (the state variables) for our material model. Now, we turn our attention to the Jacobian computation. The Jacobian has other names in literature, such as tangent stiffness, tangent modulus, or tangent operator. In the stress update algorithm, we express the deviatoric and hydrostatic parts of the stress tensor as:
The Jacobian that we want to compute is the derivative of the Second PiolaKirchhoff stress tensor with respect to the GreenLagrange strain tensor. For our example material, we assume that strains are small. This means that we do not need to distinguish between various measures of stresses and strains, because they are indistinguishable in the smallstrain limit. The derivative of stress with respect to strain is written as
If we use the equations for the deviatoric and hydrostatic stress and the definition of the trial stress, we can express the Jacobian in the following way:
Note that we replaced the increment of the plastic strain tensor by the total plastic strain tensor in the expression above. Their derivatives with respect to strain are the same, by virtue of the additive update of the plastic strains. Recall that our two modeling approaches require differently defined Jacobians. We see immediately how they are related. In the General stressstrain relation, the Jacobian is given by the full expression above. In the Inelastic residual strain, the Jacobian is given by one term in the expression, namely:
The term is the elastic Jacobian. For a purely elastic computation, the total Jacobian of the General stressstrain relation equals this quantity, while the Jacobian of the Inelastic residual strain in this case is zero.
If we use the flow rule and the chain rule for differentiation, we arrive at the following expression:
Notice that this expression depends on the plastic multiplier. This suggests that for the current material model, there is little benefit in choosing the Inelastic residual strain over the General stressstrain relation, because both approaches require a full stress update algorithm to compute the plastic multiplier. For other material models, such as creep models, the benefit would be greater. Using the governing scalar equation and the flow rule, we can compute the last derivative in the expression above.
In order to ensure rapid convergence of the global equation solver and ultimately reduce the simulation time, the computed Jacobian should be accurate. Well, what does accurate mean? Simply put, it means that the computed derivative must be consistent with the stress update algorithm that was used to compute stresses. That is, any assumptions or simplifications used in the stress update algorithm should be reflected in the computation of the Jacobian. A derivative based on the stress update algorithm is often called algorithmic or consistent.
In some situations, the Jacobian computation can be cumbersome. It is often possible in these situations to use an approximate Jacobian. Keep in mind that the accuracy of the solution is determined by the stress update algorithm. As long as the Jacobian is not too far off, the global equation solver will still converge to the correct solution, albeit at a lower rate of convergence.
In the sections above, we developed a stress update algorithm and outlined how to compute a Jacobian. Now, we will consider a special case for the hardening curve. We assume that the yield stress is a linear function of effective plastic strain. This is usually called linear hardening, and it is defined by a constant “plastic modulus” , which is the constant slope of the hardening curve. As it turns out, linear hardening means that the plastic strain increment can be solved on closed form:
In the example’s source code file, we have made use of this specialization into linear hardening.
Let’s consider an example problem of pulling a plate with a hole.
Dimensions, boundary conditions, and loads for a plate with a hole.
The problem has two symmetry planes and only onequarter of the plate is modeled. We use the following material parameters:
The problem assumes plane stress. We can compare predictions of the implementation of our material model with the builtin counterpart. We expect differences only within the order of numerical roundoff as long as the tolerance when solving the nonlinear equations is tight enough.
Computations of the effective von Mises stress in MPa of the external material implementation (left) and builtin material model (right).
Computations of the effective plastic strain for the external material implementation (left) and builtin material model (right).
There are actually more scenarios where you can employ the possibility to add external materials.
Consider a situation where you have a source code for a material model that has been verified in another context. You may have created it yourself or found it in a textbook or journal paper. In this case, it may be more efficient to use the external material functionality than casting it into a new form and enter it as set of ODEs. Even when the code is written in, for example, Fortran or C++, it is usually rather straightforward to wrap it into the C interface used by the external material.
A coded implementation may be computationally more efficient than using the User Defined or extra PDE options. The reason is that the detailed knowledge about the material law makes it possible to devise efficient stress updates, using, for example, local time stepping.
You may want to distribute your material model in a compiled form so that the end user cannot access the source code. As a matter of fact, the thirdparty product PolyUMod is implemented this way.
PolyUMod software is developed by Veryst Engineering LLC. COMSOL AB and its subsidiaries and products are not affiliated with, endorsed by, sponsored by, or supported by Veryst Engineering LLC.
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A trebuchet is a longrange weapon that uses a swinging arm to send a projectile toward a target. The machine is generally associated with hurling boulders at a castle wall to bring it down, but trebuchets have also been used to throw Greek fire and wreak all kinds of havoc. Trebuchets have appeared in several films and TV shows, such as The Return of the King (2003); Marco Polo (2014–2016); and even in Monty Python and the Holy Grail (1975), where a cow was catapulted from inside the castle walls toward an unsuspecting King Arthur!
One nonfictional and historically notable trebuchet is War Wolf (known to the English soldiers at the time as “Ludgar”). In 1304, on one of his campaigns to defeat Scotland, King Edward I besieged Stirling Castle and ordered his engineers to build a giant trebuchet. War Wolf was the largest trebuchet ever made and was rumored to send boulders of about 150 kilograms across a distance of over 200 meters.
A smallscale replica of War Wolf, a counterweight trebuchet that uses a boulderholding sling at the end of a swinging arm. Image by Ron L. Toms. Licensed under CC BY 3.0, via Wikimedia Commons.
Large trebuchets of this type would typically feature a counterweight roughly ten times the weight of the projectile, which would put War Wolf’s counterweight in the neighborhood of 1.5 tons! The poor prospects of surviving an assault from War Wolf prompted the Scottish garrison inside the castle to offer their surrender. However, the king would not have it, as he was eager to try out his new trebuchet. He forced the Scots to remain inside the castle and restarted his siege. War Wolf proved its worth, and the rest is, as they say, history.
The working principle of a trebuchet is simple. The counterweight is raised and the trebuchet is cocked. When the trebuchet is fired, the counterweight drops, and the potential energy of the system is converted into a combination of kinetic and potential energy. The projectile undergoes a swinging motion and is released at some suitable position along its trajectory. This happens when one end of the sling slips off the tip of the swinging arm.
Here, we build a computational model of a basic trebuchet with the Multibody Dynamics Module and version 5.3 of the COMSOL Multiphysics® software.
Our model uses the following assumptions and physical dimensions:
A schematic of the counterweight trebuchet model.
As the projectile is swung around by the swinging arm, it describes a nontrivial motion of varying velocity. If the trebuchet is to be designed for maximum throwing distance, a question arises: At what point during its trajectory should the projectile be released? Elementary mechanics tells us that if we neglect air resistance and the height from the ground at which the projectile is released, the throwing distance s of the projectile (measured in the positive x direction) can be expressed as
where v_{0} and α are the velocity and angle at the time of projectile release, respectively, and g is the gravitational acceleration.
Thus, finding the maximum throwing distance is equivalent to finding the combination of v_{0} and α that maximizes s. Intuitively, you might think that the angle of release should be α = 45°. Let’s see if this holds true for the trebuchet model.
The animation below shows the motion of the trebuchet as it is fired. The quantity s is shown along the projectile trajectory, and it represents the throwing distance that would follow from releasing the projectile at a certain point on this trajectory.
In the results below, the throwing distance is plotted as a function of the release angle α. The maximum throwing distance is obtained if the projectile is released at α ≈ 38°. The plot reveals that deviations of the order of 5° from this optimum only affect the throwing distance by a few meters. In other words, as long as the release angle is roughly correct, the trebuchet will function as intended.
Now, let’s examine what happens if we modify the length of the sling by ±10% using a parametric sweep. The plot below shows that the maximum throwing distance that can be obtained is greatly affected by the length of the sling. So, if you are in the business of designing trebuchets for medieval kings, you should pay attention to this design parameter.
Using a parametric sweep, you could easily examine the effect of changing other physical lengths in the model (while keeping the counterweight at fixed height for consistency). Try for yourself by downloading the model file from our Application Gallery.
In this blog post, we demonstrated that the Multibody Dynamics Module can be used to build a simple model of a counterweight trebuchet. If you are interested in learning more about multibody dynamics modeling, check out these additional blog posts:
Wondering why War Wolf was also called Ludgar? Apparently, the French name Loup de Guerre (“wolf of war”) proved more than a mouthful for the English soldiers, so it was condensed into “Ludgar”.
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Two topics frequently pop up on the COMSOL Blog: 3D printing and metamaterials. Their potential applications, such as generating customized medical implants, printing houses, and being used for cloaking technology, could transform the world around us.
A 3D printer. Image by Jonathan Juursema — Own work. Licensed under CC BYSA 3.0, via Wikimedia Commons.
By combining these two technologies, we might be able to use directlaserwriting (DLW) printing to fabricate complicated metamaterials, a process that could be challenging or impossible with other manufacturing techniques. Using this idea as inspiration, a team from the Karlsruhe Institute of Technology (Germany) and Université de Bourgogne FrancheComté (France) investigated a metamaterial that displays the unique mechanical property of negative effective compressibility under stable and static conditions.
The researchers’ poroelastic metamaterial is a 3D manmade composite that experiences isotropic expansion when the hydrostatic pressure generated by the surrounding environment is increased. Most natural elastic materials react in the opposite way, reducing their volume when exposed to an increased hydrostatic pressure at a fixed temperature.
A sponge is a material that is affected by poroelastic phenomena.
So why does this metamaterial expand? To answer this, let’s take a look inside the metamaterial, which consists of a single ordinary constituent solid. Within the material are hollow 3D crosses with concealed internal volumes that contain air at a constant pressure. Each cross also has circular membranes attached to its ends.
These membranes warp inward or outward when the surrounding pressure is different than the pressure within the crosses. The bars asymmetrically connected to the membranes translate this warping into a cross rotation. If the hydrostatic pressure outside is greater than the pressure inside, then the individual rotations translate into an isotropic expansion of the structure, causing a negative effective compressibility.
A unit cell at zero pressure (left) and an elevated pressure (right), depicting the principle of negative compressibility. Image courtesy Jingyuan Qu and Muamer Kadic.
While this negative compressibility may appear to violate the laws of physics, the effective volume increase corresponds with an unseen volume decrease within the material. This ensures that the structure is stable.
To examine the detailed structure of the innovative metamaterial, the researchers turned to the COMSOL Multiphysics® software. When asked about the benefits of this approach, Jingyuan Qu — a member of the research team — noted how easy it was to implement.
The model of the metamaterial is a single unit cell. To see what happens when there is a difference between the pressure inside and outside the material, a pressure increase is implemented as a normal force on all of the outer surfaces of the model. Further, the model is simulated under periodic boundary conditions, enabling the researchers to successfully find the effective material parameters.
Note that builtin periodic boundary conditions are available in the Structural Mechanics and MEMS modules.
For their research, the team performed two main numerical experiments:
In their experiments, the team used the equations for standard linear elasticity:
Now, let’s take a look at the second numerical experiment.
To mimic an infinite material case, periodic conditions are implemented so that each side of the unit cell must contract or expand isotropically. First, selections for each side of the structure are created and named according to the directions x+, x, y+, y, z+, and z. Then, probe variables were created, giving the average displacement on the “minus” sides (dispx, dispy, dispz), shown in the second screenshot below.
Examples for the x direction case, displaying the process of selecting the boundaries that connect to the next unit cell, shown for one of the six planes (top) and the boundary probe settings (bottom). Images courtesy Jingyuan Qu and Muamer Kadic.
Next, the probe variables are used as boundary conditions on both sides (prescribed displacement). That is, on the ‘x’ boundaries, the x direction displacement is set to dispx, while on the ‘x+’ boundaries, it is set to dispx. Similar boundary conditions are then set on the other periodic cuts. The idea is that the displacement dispx, still unknown, becomes part of the solution. Since the conjugate reaction force to the prescribed displacements must be zero, the structure will expand or contract in such a way that there is no net force.
Prescribing the probed displacement. Image courtesy Jingyuan Qu and Muamer Kadic.
Moving on, outer pressure is also applied. After selecting the outer boundaries of the geometry and using a high angular tolerance, the model shows that the inner boundaries in the concealed volumes are not selected, as seen below.
The outer boundary settings. Image courtesy Jingyuan Qu and Muamer Kadic.
The hydrostatic load is then applied as a boundary load with pressure (P).
The hydrostatic pressure is implemented as a normal force acting on all outer boundaries. Image courtesy Jingyuan Qu and Muamer Kadic.
The resulting structure of the poroelastic metamaterial at different angles. Images courtesy Jingyuan Qu.
As a point of comparison, the researchers also examined an ordinary porous structure and a cube of a continuous isotropic material. When exposed to an increased hydrostatic pressure, both of these structures shrink in volume. Under the same conditions, the porous metamaterial expands — highlighting its negative effective compressibility.
Thanks to their extensive research, the team was able to capture the metamaterial’s behavior, improve its design, and use this information to move on to the fabrication stage. While fabricating this material may not be possible with conventional machining techniques, 3D printing serves as an alternative option for creating negative compressibility metamaterials. 3D printers can form this metamaterial by using ordinary materials that shrink under hydrostatic pressure.
Qu notes that after it is realized, the metamaterial may find uses in highpressure applications due to its ability to maintain a constant effective volume, even when subjected to highpressure environments.