Phononic crystals are rather unique materials that can be engineered with a particular band gap. As the demand for these materials continues to grow, so does the interest in simulating them, specifically to optimize their band gaps. COMSOL Multiphysics, as we’ll show you here, can be used to perform such studies.

A *phononic crystal* is an artificially manufactured structure, or material, with periodic constitutive or geometric properties that are designed to influence the characteristics of mechanical wave propagation. When engineering these crystals, it is possible to isolate vibration within a certain frequency range. Vibration within this selected frequency range, referred to as the *band gap*, is attenuated by a mechanism of wave interferences within the periodic system. Such behavior is similar to that of a more widely known nanostructure that is used in semiconductor applications: a *photonic crystal*.

Optimizing the band gap of a phononic crystal can be challenging. We at Veryst Engineering have found COMSOL Multiphysics to be a valuable tool in helping to address such difficulties.

When it comes to creating a band gap in a periodic structure, one way to do so is to use a unit cell composed of a stiff inner core and a softer outer matrix material. This configuration is shown in the figure below.

*A schematic of a unit cell. The cell is composed of a stiff core material and a softer outer matrix material.*

Evaluating the frequency response of a phononic crystal simply requires an analysis of the periodic unit cell, with Bloch periodic boundary conditions spanning a range of wave vectors. It is sufficient to span a relatively small range of wave vectors covering the edges of the so-called *irreducible Brillouin zone* (IBZ). For rectangular 2D structures, the IBZ (shown below) spans from Γ to X to M and then back to Γ.

*The irreducible Brillouin zone for 2D square periodic structures.*

The Bloch boundary conditions (known as the Floquet boundary conditions in 1D), which constrain the boundary displacements of the periodic structure, are as follows:

u_{destination} = exp[-i\pmb{k}_{F} \cdot (r_{destination} - r_{source})] u_{source}

where **k**_{F} is the wave vector.

The source and destination are applied once to the left and right edges of the unit cell and once to the top and bottom edges. This type of boundary condition is available in COMSOL Multiphysics. Due to the nature of the boundary conditions, a complex eigensolver is needed. The system of equations, however, is Hermitian. As such, the resulting eigenvalues are real, assuming that no damping is incorporated into the model. The COMSOL software makes this step rather easy, as it automatically handles the calculation.

We set up our eigensolver analysis as a parametric sweep involving one parameter, *k*, which varies from 0 to 3. Here, 0 to 1 defines a wave number spanning the Γ-X edge, 1 to 2 defines a wave number spanning the X-M edge, and 2 to 3 defines a wave number spanning the diagonal M-Γ edge of the IBZ. For each parameter, we solve for the lowest natural frequencies. We then plot the wave propagation frequencies at each value of *k*. A band gap appears in the plot as a region in which no wave propagation branches exist. Aside from very complex unit cell models, completing the analysis takes just a few minutes. We can therefore conclude that this approach is an efficient technique for optimization if you are targeting a certain band gap location or if you want to maximize band gap width.

To illustrate such an application, we model the periodic structure shown above, with a unit cell size of 1 cm × 1 cm and a core material size of 4 mm × 4 mm. The matrix material features a modulus of 2 GPa and a density of 1000 kg/m^{3}. The core material, meanwhile, has a modulus of 200 GPa and a density of 8000 kg/m^{3}. The figure below shows no wave propagation frequencies in the range of 60 to 72 kHz.

*The frequency band diagram for selected unit cell parameters.*

To demonstrate the use of the band gap concept for vibration isolation, we simulate a structure consisting of 11 x 11 cells from the periodic structure analyzed above. These cells are subjected to an excitation frequency of 67.5 kHz (in the band gap).

*The structure used to illustrate vibration isolation for an applied frequency in the band gap.*

The animation below highlights the response of the cells. From the results, we can gather how effective the periodic structure is at isolating the rest of the structure from the applied vibrations. The vibration isolation is still practically efficient, even if fewer periodic cells are used.

*An animation of the vibration response at 67.5 kHz.*

Note that at frequencies outside of the band gap, the periodic structure does not isolate the vibrations. These responses are depicted in the figures below.

*The vibration response at frequencies outside of the band gap. Left: 27 kHz. Right: 88 kHz.*

To learn more about the 2D band gap model presented here, head over to the COMSOL Exchange, where it is available for download.

- P. Deymier (Editor),
*Acoustic Metamaterials and Phononic Crystals*, Springer, 2013. - M. Hussein, M. Leamy, and M. Ruzzene,
*Dynamics of Phononic Materials and Structures: Historical Origins, Recent Progress, and Future Outlook*, Appl. Mech. Rev 66(4), 2014.

Nagi Elabbasi, PhD, is a managing engineer at Veryst Engineering LLC. Nagi’s primary area of expertise is the modeling and simulation of multiphysics systems. He has extensive experience in the finite element modeling of structural, CFD, heat transfer, and coupled systems, including fluid-structure interaction, conjugate heat transfer, and structural-acoustic coupling. Veryst Engineering provides services in product development, failure analysis, and material testing and modeling, and is a COMSOL Certified Consultant.

]]>Using COMSOL Multiphysics, we implemented a wear model and validated it by simulating a pin-on-disc wear test. We then used the model to predict wear in an automotive disc brake problem. The results we found showed good agreement with published wear data.

*Wear* is the process of the gradual removal of material from solid surfaces that are subjected to sliding contact. It is a complex phenomenon that is relevant to many problems involving frictional contact, such as mechanical brakes, seals, metal forming, and orthopedic implants. The rate of wear depends on the properties of the contacting materials and operating conditions.

Archard’s law is a simple but widely used wear law that relates the volume of material removed due to wear W to the normal contact force F_N, sliding distance L_T, material hardness H, and a material-related constant K

W=\frac{KF_N L_T}{H}

In our work, we considered a modified version of Archard’s law:

\.{w}=k(H,T)p_N V_T

This modified law relates the wear depth w at any point to the normal contact pressure p_N, magnitude of sliding velocity V_T, and a constant k that is a function of the material and temperature. The wear constant k may be computed from experimental wear data, which is typically in the form of weight loss for a specific contact pressure and velocity.

Wear equations are not directly available in finite element analysis (FEA) codes, although their implementation in COMSOL Multiphysics is straightforward. We incorporated the wear equations within our simulations by defining boundary ordinary differential equations (ODEs) on the destination contact surfaces with the wear depth w as the independent variable. The wear depth w is then used as an offset between contacting surfaces (e.g., brake pad and disc) within the contact formulation in COMSOL Multiphysics. In particular, contact is enforced when the penetration between the contact surfaces is equal to the wear depth w, as shown in image below.

*Modification of contact gap calculation: w is the wear depth, g is the gap, and \lambda is the contact pressure.*

This wear algorithm is very efficient since it does not involve altering the nodal locations to account for material loss due to wear. It is only suitable, however, for cases where the wear depth is significantly less than the width of the contact surface.

You can enhance this wear algorithm by including more sophisticated effects, such as anisotropic wear behavior, dependence on the mean and deviatoric stresses in the solid (not just the contact pressure), threshold pressure/stress below which no wear occurs, and more. The assumption of small wear depth must still hold for this modeling approach to be accurate.

We validated the new, contact-offset-based wear model implementation by simulating a pin-on-disc wear test. Only a small section of the disk is modeled, as shown below.

*Pin-on-disc wear test model.*

The disc in this model is much stiffer than the pin and all the wear is assumed to occur in the pin. A force is applied to the pin, resulting in a circular, Hertzian-type contact pressure distribution. A constant tangential velocity is then applied to the disc. The graph below shows how the wear depth varies radially along the pin at four time instances. The total volume loss, calculated as the integral of wear depth over the contact surface, was similar to the value calculated using Archard’s law.

*Wear depth vs. radial distance in the pin-on-disc model.*

We also used the model to predict wear in an automotive disc brake problem, which is similar to the Heat Generation in a Disc Brake model that can be downloaded from the COMSOL Model Gallery. We developed a 3D thermal-structural disc brake model involving representative brake disc/rotor and brake pads.

*Disc brake model used in the COMSOL Multiphysics wear simulation.*

The structural and thermal processes are coupled through frictional heat generation, thermal expansion, and thermal contact. Both physics fields are also coupled to the wear depth evolution boundary ODE. We used a fully-coupled direct solver that converged rapidly, keeping solution times similar for problems with and without wear.

The results for both the pin-on-disc validation example and the disc brake problem were in good agreement with published wear data. In the disc brake example, the model captured the non-uniform wear rate that is typically observed on brake pads; it was higher near the outer radius and leading edge, as shown below.

*Typical brake pad wear depth profile.*

We will present more of our results, including contact pressure and wear contours, at both the Cambridge and Boston stops of the COMSOL Conference 2014.

Nagi Elabbasi, PhD, is a Managing Engineer at Veryst Engineering LLC. Nagi’s primary area of expertise is modeling and simulation of multiphysics systems. He has extensive experience in finite element modeling of structural, CFD, heat transfer, and coupled systems, including fluid-structure interaction, conjugate heat transfer, and structural-acoustic coupling. Veryst Engineering provides services in product development, material testing and modeling, and failure analysis, and is a member of the COMSOL Certified Consultant program.

]]>When thin structures such as beams, plates, or shells are immersed in a fluid, their natural frequencies are reduced. The fluid also affects their mode shapes and is a source of damping. This phenomenon affects structures across a wide range of industries and sizes, from micro-scale structures (e.g. MEMS actuators) to larger structures (e.g. ships).

Today, we will take a look at a model of a cantilever beam immersed in a fluid:

An approximate analytical solution of the form shown below is frequently used to estimate the natural frequencies of the immersed beam. This is estimated based on the *structure-only* natural frequencies, beam geometry, and the ratio of fluid-to-beam densities. The analytical expression approximately accounts for the added mass of the fluid that is displaced by the beam. It does not account for viscous effects.

We at Veryst Engineering used COMSOL Multiphysics to determine the natural frequencies and mode shapes of an immersed cantilever beam. Then, we compared the results with the analytical approximation.

We set up the problem as a coupled acoustic-structure eigenvalue analysis. To account for the mass of the fluid, we selected a pressure acoustics formulation, and we accounted for damping due to fluid viscosity by including a viscous loss term. We assumed the fluid space to be sealed. The COMSOL software automatically detects the solid-fluid boundary and applies the necessary boundary conditions at the solid-fluid interface.

Below, you can see a table with the first and fourth natural frequencies (in kHz) of beams in vacuum, air, and water:

As expected, the results show that air has a minor effect on the beam, while water reduces the lowest natural frequencies of the beam by about 20%. Also shown in the table is the analytical estimate for a beam immersed in water. The analytical estimate is close to the COMSOL Multiphysics prediction for this relatively standard beam configuration.

Next, we can have a look at a couple of animations of our results.

The first animation depicts the fourth mode of deformation for a beam immersed in water:

*Fourth mode shape of cantilever beam immersed in water.*

The second animation demonstrates fluid pressure contours and fluid velocity arrow plots at a section along the beam, again for the fourth natural frequency of the beam.

*Velocity and pressure contours for fourth beam natural frequency.*

This modeling example involved a simple cantilever to illustrate the concept. However, the coupled structural-acoustic modeling approach used is also applicable to more realistic geometries, such as ship hulls and MEMS actuators.

Nagi Elabbasi, PhD, is a Managing Engineer at Veryst Engineering LLC. Nagi’s primary area of expertise is modeling and simulation of multiphysics systems. He has extensive experience in finite element modeling of structural, CFD, heat transfer, and coupled systems, including fluid-structure interaction, conjugate heat transfer, and structural-acoustic coupling. Veryst Engineering provides services in product development, material testing and modeling, and failure analysis, and is a member of the COMSOL Certified Consultant program.

]]>Two weeks ago I led a webinar on fluid-structure interaction (FSI) using COMSOL Multiphysics. FSI involves coupling between a deformable or moving structure and a surrounding or internal fluid flow. There is a growing number of engineering and scientific problems where a purely structural or purely CFD analysis just aren’t accurate enough. Both analyses have to be accounted for simultaneously. Some examples are valve chatter, damping in MEMS, cardiovascular modeling, and shock absorbers.

The webinar highlighted the important features and capabilities available for FSI analysis in COMSOL Multiphysics, including the types of FSI coupling, the solution algorithms used, the available solvers, and the handling of the moving fluid mesh. COMSOL provides a wide range of capabilities for the advanced user, and also automates most of the steps required for FSI analysis, which is great for both novice and advanced users.

COMSOL offers two types of solvers for fluid-structure interaction problems (as well as other multiphysics problems). The first is the fully coupled solver, or *monolithic* solver as it is sometimes called in literature, and the second is the segregated solver (or *partitioned* solver). Having both solvers enables optimal solver selection for a wide range of FSI problems. The default solver settings work well for most problems, but there are also a lot of solver functionalities for advanced users to adjust for tougher problems. The fluid mesh movement algorithm can also handle severe mesh deformation, as shown below.

We also demonstrated how we at Veryst Engineering used COMSOL Multiphysics to set up two “non-standard” FSI problems and make engineering simplifications that significantly reduce solution times. In the case of a sea floor energy harvester, the solid part of the model was reduced to one degree of freedom (vane rotation), and in the case of the stretching of a fluid-filled hose, the enclosed fluid region was reduced to a single global constraint.

After the two non-standard FSI problems, we showed an interesting FSI example involving a 3D peristaltic pump. Peristaltic pumps move fluid by squeezing on a tube, causing the fluid inside to move. The deformation of the tube is strongly coupled to the fluid flow inside it, so an FSI analysis is required. The model is also nonlinear due to the large rotations of the roller, the nonlinear material response of the tubes, and the contact between the rollers and the tube. We were able to use this model to predict stresses in the tube and flow conditions, including flow fluctuations. The figure below shows the von Mises stresses in one tube configuration. You can find more details on this pump model in a paper we presented at the COMSOL Conference 2011.

We also held a live demo, showing the important steps involved in setting up and running a fluid-structure interaction analysis in COMSOL. We set up a 2D simplified version of the peristaltic pump model that captures some of the flow characteristics of the full 3D model.

Interested in learning more about FSI analyses? There is an archived version of the FSI webinar available for your viewing.

Nagi Elabbasi, PhD, is a Managing Engineer at Veryst Engineering LLC. Nagi’s primary area of expertise is modeling and simulation of multiphysics systems. He has extensive experience in finite element modeling of structural, CFD, heat transfer, and coupled systems, including fluid-structure interaction, conjugate heat transfer, and structural-acoustic coupling. Veryst Engineering provides services in product development, material testing and modeling, and failure analysis, and is a member of the COMSOL Certified Consultant program.

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