As we just learned, the fully coupled approach to solving a steady-state nonlinear problem actually uses the exact same damped Newton-Raphson algorithm used to solve a single physics nonlinear problem. Although this algorithm does converge well for many cases, it can fail or converge very slowly if the choice of initial conditions are poor. It should come as no surprise then that the techniques we have already looked at, such as Load Ramping and Nonlinearity Ramping, are just as valid when applied to a multiphysics problem. In fact, there is really nothing to add to these techniques — they can be used equivalently.
There is one new variation of the nonlinearity ramping technique, and that is to ramp the coupling between the physics. Numerically, it is in fact identical to the nonlinearity ramping technique already discussed, but conceptually it is the magnitude of the couplings between the physics that is ramped up, rather than the magnitude of the nonlinearity in a single physics. The only difficulty is choosing, and implementing, the term that should be ramped. Luckily, most multiphysics problems have quite obvious couplings between the physics, which can be found simply by writing out the governing equations and boundary conditions and examining how the material properties and loads are dependent upon the variables being solved for.
The most important thing to remember is that the underlying algorithm used to solve a fully coupled multiphysics problem is exactly the same as the algorithm used to solve a nonlinear single physics problem. Keeping this in mind, you will find that fully coupled mutliphysics problems really do not pose any additional conceptual hurdles beyond understanding how the physics in the model interact with each other.
On the other hand, the segregated approach can lead to a variety of different solution strategies that can greatly accelerate solution convergence, and significantly affect the amount of memory needed to solve the problem. To understand this, let’s make a flowchart of the different multiphysics solution techniques. Consider the same problem from our previous blog post about a busbar that heats up due to current flow and experiences thermal stresses.
First, the fully coupled solver starts from an initial guess and applies Newton-Raphson iterations until the solution has converged:
When solving such a problem, you will get a Convergence Plot, which shows the error estimate decreasing between Newton-Raphson iterations. Ideally, the error should go down monotonically if it does converge, then start investigating ramping the loads, the nonlinearities, or the multiphysics couplings. This approach will almost always require a more memory-intensive direct solver to solve the linear system of equations in each Newton-Raphson step.
Now, compare the fully coupled approach to the segregated approach, which solves each physics sequentially until convergence:
You will get a different kind of convergence plot for such a problem, one that shows the error associated with each physics you are solving. Each of the physics can use the optimal solver, either the direct or the less memory-intensive iterative, to solve the linear system of equations. Each segregated step can be a nonlinear problem on its own, and can be solved to a desired tolerance, and with custom damping, as appropriate for the particular combination of physics problem being solved.
With this solution method, you will get at least two convergence plots, one for the iterative solver(s) possibly used within a segregated step, and a second for the overall convergence of the segregated approach:
The above plot shows the decrease in error for each physics. Although more iterations may be required for the same problem, each loop through the segregated solution approach can be much faster than the Newton-Raphson step required for the fully coupled approach. You can also get a little bit more information out of this, if only one or two physics are not converging, then you will want to check the set-up of these first.
One thing you may recall about this problem is that the temperature change is driven by the resistive heating from the current, and the current distribution depends upon the electrical conductivity, which is temperature-dependent. That is, the voltage and temperature solutions are bi-directionally coupled. On the other hand, although the thermal strain and the Young’s Modulus are dependent upon temperature, the voltage and temperature solutions do not depend upon the displacements or stresses. That is, there is a uni-directional coupling from the thermal problem to the structural problem. We can immediately see that there is an even more efficient way to solve this problem. We can solve the voltage and temperature problem first and subsequently solve for the displacements:
So, we can see that there are (at least) three different ways of solving this problem: fully coupled, segregated and assuming couplings between all of the physics are segregated, or with a sequential solution step to take advantage of the uni-directional coupling between temperature and displacements. When solving a multiphysics problem, COMSOL will assume coupling between all physics, and try to choose the optimal fully coupled or segregated approach, based on the physics and the problem size. Of course, it is always instructive to go into the solver settings to see what settings the software has chosen.
This series of postings has been designed to give you an understanding of the algorithms used in COMSOL to solve single physics and multiphysics linear and nonlinear steady-state problems. Issues such as meshing, accuracy, and convergence have been covered. With this information, you should be able to more confidently address the solutions to your models of this type.
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Let’s start by considering a very simple steady-state multiphysics problem: A coupling of steady-state electric current flow through a metal busbar, heat transfer in the bar, and structural deformations. Resistive heating arises due to the current flow, which raises the temperature of the bar and causes it to expand. In addition, the temperature rise will be significant enough that the electrical, thermal, and structural material property variations with temperature must be considered. We want to find the current flow, temperature fields, and deformations and stresses under steady-state conditions. The figure below shows a schematic of the problem being solved.
The multiphysics problem at hand.
There are here three governing partial differential equations being solved. First off, the equation that describes the voltage distribution within the domain is:
After discretizing via the finite element method, we can write a system of equations as:
where the subscript, , denotes the voltage unknowns, and the system matrix, , is dependent upon the temperature unknowns, . Assuming that the voltage distribution is known, then the volumetric resistive heating can be computed from:
where , the electric field, is: . This heat source shows up in the governing equation for temperature:
And this equation gives us the system of equations:
Once we have the temperature distribution within the domain, we can solve for the structural displacements:
where the elasticity matrix, , is computed based on the temperature-dependent Young’s Modulus, . The imposed thermal strain is and the strain is . The system of equations that solve for the displacements is written as:
where the subscript, , indicates the displacement unknowns.
We can combine these systems of equations together:
We can see by examination that this is a nonlinear problem, and as we learned earlier, this requires that we find the solution by taking Newton-Raphson iterations until we get convergence:
That is really all there is to it! There is no conceptual difference at all between solving a single physics nonlinear problem and solving a coupled physics problem. Everything that we have already learned about solving nonlinear single physics problems, including all of the discussions about damping, load and nonlinearity ramping, as well as meshing, is just as valid for solving a multiphysics problem.
But it is also important to understand a (sometimes very serious) drawback to the above approach. During the Newton-Raphson iteration, we need to evaluate the derivative, , so let’s write that out:
where the comma derivative notation is used, e.g.: .
Clearly, the above matrix is non-symmetric, and this can lead to a problem: If the system matrix is not definite, then we may need to use the more memory-intensive direct solvers. (Although iterative solvers, with the right choice of preconditioner, can solve a wider class of problems they cannot be guaranteed to handle all cases.) Solving such a multiphysics problem with a direct solver will be both memory- and time-intensive.
However, there is an alternative. The above method, called a Fully Coupled approach, assumes that all of the couplings between the physics must be considered at the same time. In fact, for the purposes of solving many types of multiphysics problems, we can neglect these off-diagonal terms during the solution, and solve using a more memory- and time-efficient Segregated approach.
The Segregated approach treats each physics sequentially, using the results of the previously solved physics to evaluate the loads and material properties for the next physics being solved. So, using the above example, we first take a Newton-Raphson iteration for the voltage solution:
where, for the first iteration, we must have a starting guess for voltage and temperature . The material properties are evaluated using the initial conditions given for the temperature field. Next, the temperature solution is evaluated:
where, for the first iteration, , the initial conditions given for the temperature field are used to evaluate the materials properties, , but the loads are evaluated based upon the voltage solution that was just computed: . Similarly, the displacement field is solved:
where the material properties and loads for the structural problem are computed using the temperature field computed above.
These iterations are then continued: voltage, temperature, and displacement are repeatedly computed in sequence. The algorithm is continued until convergence is achieved, as defined earlier.
The great advantage to the above approach is that the optimal iterative solver can be used in each linear substep. Not only are you now solving a smaller problem in each substep, but you can also use a solver that is more memory-efficient and generally solves faster. Although the segregated approach generally does require more iterations until convergence, each iteration takes significantly less time than one iteration of the fully coupled approach.
The algorithm used by the segregated solver for a model composed of n number of different physics is:
For general multiphysics problems, you will still have to choose the order in which the physics are solved, but the software has default suggestions as to an appropriate sequence for all built-in multiphysics interfaces. COMSOL Multiphysics will provide default linear solver settings for each physics in the segregated sequence.
When the segregated approach is applicable, it will converge to the same answer as the fully coupled approach. The segregated approach will usually take more iterations to converge; however, the memory and time requirements for each sub-step will be lower, so the total solution time and memory usage can be lower with the segregated approach.
In this blog post, we have outlined the two classes of algorithms used to solve multiphysics problems — the Fully Coupled and the Segregated approach. The Fully Coupled approach is essentially identical to the Newton-Raphson method already developed for solving single physics nonlinear problems. It was shown to be very memory-intensive, but is useful, and generally needed, for multiphysics problems that have very strong interactions between the various physics being solved. On the other hand, the Segregated approach assumes that each physics can be solved independently, and will iterate through the various physics in the model until convergence.
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There are three key points that you should recall from the blog post on meshing considerations for linear static problems. These are:
When addressing nonlinear problems, we have already learned that even a finite element problem with a single degree of freedom may not converge, even for a problem that has a solution. We have learned several techniques that can address this issue, but have not yet introduced the interplay of the mesh and the nonlinear solver.
The single most important thing to keep in mind when meshing nonlinear problems is this:
Even if the problem is well-posed, and even if we have chosen a good solution method, the problem may still fail to converge if the problem is not meshed finely enough in the regions of strong nonlinearities.
To understand this, let’s take a look at a one-dimensional thermal finite element problem. We will consider a 1 m thick wall with a fixed temperature of at one end at at the other, as shown below:
We will examine the solutions to this problem for the different thermal conductivities plotted below:
If we plot out the solution for the linear case, , we get:
By examination, we see that the solution is a straight line. For this case, the solution can be found by using a single linear element across the entire domain.
Now, if we plot out the case , with elements delineated by dashed lines, we get:
We can see that the solution to this nonlinear problem will require more than a single element across the domain. In fact, regardless of how many elements we use, the polynomial basis function will never perfectly match the true solution. We can successively refine the mesh everywhere in the domain and get closer and closer to the true solution, just as we did for a linear problem.
Finally, if we plot out the case , we get:
This solution is more complicated. There are clearly regions of the solution where a single element would be almost sufficient to completely describe the solution. Yet, there are regions where the solution varies quite rapidly as a function of position. These regions are around , where there are strong nonlinearities in the material property function. Although the material property function has only one region of strong nonlinearity with respect to temperature, the solution exhibits two regions over the domain where the solution varies rapidly. Only these regions in space require a finer mesh. In fact, the solver might not converge at all if the mesh in these regions is too coarse.
For these types of problems, adaptive mesh refinement becomes highly motivated, since the locations of the gradients in the modeling domain are generally not known ahead of time. Ramping of the nonlinearities in the model is also helpful, since starting with a linear problem will result in a problem that can always be solved, regardless of the mesh. By gradually ramping up the nonlinearity, and performing adaptive mesh refinement iteratively, it is possible to improve model convergence for nonlinear problems.
Meshing of nonlinear stationary finite element problems is inherently linked with the question of getting a nonlinear model to converge. Convergence rates, and even the possibility of convergence, are dependent on both the solver algorithm used and the mesh. All of the techniques mentioned up to this point: manual and adaptive mesh refinement, choosing of initial conditions, load ramping, nonlinearity ramping, and any combination of these techniques may be needed as you develop more and more sophisticated models. Finally, always keep in mind that a mesh refinement study is needed to assess solution accuracy.
For an example model that incorporates all of the techniques that we have learned about thus far, please see the Cooling and Solidification of Metal model. Mastering these techniques will allow you to quickly and efficiently model nonlinear problems.
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Consider again the system shown below, of a force applied to a spring with nonlinear stiffness.
We’ve seen that we can solve this problem using the Newton method with damping or by using the continuation method and ramping the load to give the Newton method good starting points. Now we’ll examine how to ramp the nonlinearity. First let’s take another look at the function describing the force balance on our single node:
We can re-write this more generally as: where is the nonlinear spring stiffness. Now we can solve a different problem that uses a stiffness defined as:
In other words, we divide our spring stiffness function into two parts — a linear term, , and a nonlinear term, — and then introduce an additional parameter, , that interpolates between the linear and nonlinear case. We then use the same Newton method as before on a series of problems with the parameter , getting ramped from zero to one. That is, we use the continuation method to ramp from a (simple to solve) linear problem to a (more difficult) nonlinear problem.
Next we will look at solving the above example by using this technique. Our original spring stiffness, , gets re-written as:
We start by solving for and get a linear spring stiffness of , so now all we need to do is choose a linearization point, . For this example, if we choose , we see that . Recall our discussion about solving linear static finite element problems, where we learned that you will always find the solution to a linear problem in a single Newton iteration. Now, ramp up the parameter , as shown:
Clearly, only a few Newton iterations, starting from the solution to the case, are needed to solve . So we can repeat with and thereby ramp from the fully linear to the fully nonlinear case.
This method is attractive because you can always find a solution to a linear problem, so you can always solve for . You only need to consider which point to linearize about initially and what kind of nonlinearity ramping to use.
We can also use the concept of nonlinearity ramping to address the case where the nonlinear terms are not continuously differentiable. Recall the case from the blog post on Solving Nonlinear Static Finite Element Problems, the system with the piecewise constant spring stiffness, for , for , and for , would result in a force balance function:
As we saw earlier, this problem cannot be solved by the Newton method, unless you happen to start within the (very small) radius of convergence of the solution. But now consider replacing the original spring stiffness with a smoothed stiffness that can be ramped up as shown in the figure below:
Clearly, this problem is solvable, and we can use this technique to get an approximate solution to the original problem. Using this method just requires that we find an appropriate smoothing function and nonlinearity ramping path.
Whenever you have a problem which has the kind of stepped behavior shown above, it is also worth trying out the Double Dogleg nonlinear solver instead of the Newton method. The Double-Dogleg is a Trust Region solver that works well when solving problems where the Newton method may oscillate between two different regions. A good physical example of this is a structural contact problem, where there is a sudden transfer of load as two objects come into physical contact.
We have now seen two methods for improving the convergence of nonlinear problems: load ramping and nonlinearity ramping. In practice, either or both methods can be used, and, through the careful design of your material properties and loads, it is possible to blend the two approaches. It can be difficult to say ahead of time which method will perform better, and each model you work on will require some experimentation in terms of the load ramping path, the nonlinearity ramping, and the choice of initial condition for linearization. Also, if you expect that the solution may oscillate between different cases, the Double-Dogleg solver can perform better than the Newton method. With experience, you will build up your engineering intuition about how best to solve the classes of problems that you are working on.
The techniques introduced here often work well for nonlinear static finite element problems where it may be difficult to find good initial conditions, or problems which have strong nonlinearities and discontinuities in the material properties. In practice, a very wide class of problems can be addressed using these approaches. However, you must also be aware that there are different meshing requirements when solving nonlinear problems. That is the next topic we will address, so stay tuned.
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Consider again the system of a force applied to a spring with nonlinear stiffness:
We can solve this problem with the damped Newton-Raphson method as long as we choose an appropriate initial condition (earlier we chose ). In the other blog entry, we noticed that choosing an initial condition outside of the radius on convergence, any point for example, will cause the solver to fail. Now, for this single degree of freedom problem we can easily determine the radius of convergence, but for typical finite element problems it would be much harder. So instead of trying to find the radius of convergence, let’s instead apply a little bit of physical intuition to this problem.
Here we are applying a load, , to a system and we are trying to find a solution by starting from an initial condition, . But what happens if we apply a load ? Newton’s First Law tells us that a system under no load will have no deformations. So what happens if we apply a load, , of magnitude infinitesimally larger than zero? It would be reasonable to assume that the Newton-Raphson method, starting from , will be able to find a solution, . It is also reasonable to say that we can then increment the load to such that and again find a solution , as long as the load increment is small enough. Repeating this algorithm, we will eventually get to the final load , and our desired solution. That is, starting from a zero load, and zero solution, we gradually ramp up the load until we achieve the desired total load. This procedure is plotted in the figure below. The dark arrows indicate where the Newton-Raphson iterations start for a particular load value.
This algorithm is also referred to as a continuation method on the load. This gradual ramping up the load from a value close to zero is often a more robust approach to solving nonlinear problems via the damped Newton method, since the previous solutions are good initial guesses for the next step.
With this algorithm, we not only have a good way of addressing the issue of finding a good starting point for the Newton-Raphson iterations, we also have an algorithm that is useful for the case of a problem that does not have a solution. Consider again the problem where the spring gets weaker as it is pulled, where as discussed previously. This problem does not have a solution. In this case, we can analytically determine that for any load there is no solution. But if we use a smaller load, then the system is stable. In fact, in our scenario the system is bi-stable; there are two solutions for every load . Although, we are probably only interested in the branch we get to starting from and . Let us plot out :
Now let’s also assume that we do not know that the peak possible load is at , and examine what happens when COMSOL tries to solve this problem for . If we plot out for we see that for there is no solution to be found. The continuation solver in COMSOL will then automatically perform a search over the interval between the last successful load value and the next desired load step. That is, the solver tries to backtrack to find an intermediate solution that can then be used as a starting value for the next step. This algorithm is always used whenever the Continuation Method feature (or the Parametric Sweep feature) is used on a single parameter when solving a stationary problem. In that case, the solver will be able to find the approximate failure load of the system, which is also very useful information.
We have now introduced the concept of load ramping and using the continuation method to improve the robustness of the Newton method. Since a system with no load has a known solution, we have seen that this technique can eliminate the question of what value to choose for the initial condition. We also learned that it is possible to approximately find the failure load. For these reasons, load ramping is one important technique that you should understand when setting up and solving nonlinear static finite element problems.
Let’s take a look at a log file from a nonlinear finite element problem. We’ll set up and solve the problem described above, of a nonlinear spring that gets weaker as we pull on it. We know that this problem does not have a solution, so lets see what happens:
Stationary Solver 1 in Solver 1 started at 15-Jul-2013 11:26:46. Parametric solver Nonlinear solver Number of degrees of freedom solved for: 1. Parameter P = 0.2. Symmetric matrices found. Scales for dependent variables: State variable u (mod1.ODE1): 1 Iter ErrEst Damping Stepsize #Res #Jac #Sol 1 0.18 1.0000000 1 2 1 2 2 0.013 1.0000000 0.22 3 2 4 3 6.5e-005 1.0000000 0.015 4 3 6 Parameter P = 0.3. Iter ErrEst Damping Stepsize #Res #Jac #Sol 1 0.025 1.0000000 0.21 7 4 9 2 0.00069 1.0000000 0.031 8 5 11 Parameter P = 0.4. Iter ErrEst Damping Stepsize #Res #Jac #Sol 1 0.89 1.0000000 2.7 11 6 14 2 0.3 0.8614583 0.76 12 7 16 3 0.2 0.8154018 0.43 13 8 18 4 0.31 0.4194888 0.42 14 9 20 5 0.86 0.0836516 0.9 15 10 22 Parameter P = 0.325. Iter ErrEst Damping Stepsize #Res #Jac #Sol 1 0.089 1.0000000 0.4 18 12 26 2 0.014 1.0000000 0.13 19 13 28 3 0.0003 1.0000000 0.018 20 14 30 Parameter P = 0.375. Iter ErrEst Damping Stepsize #Res #Jac #Sol 1 0.099 1.0000000 0.32 23 15 33 2 0.079 0.9390806 0.19 24 16 35 3 0.2 0.3028345 0.24 25 17 37 4 0.94 0.0302834 0.95 26 18 39 ... SOME PARTS OF THIS LOG FILE OMITTED ... Parameter P = 0.368359. Iter ErrEst Damping Stepsize #Res #Jac #Sol 1 0.046 1.0000000 0.057 80 49 112 2 0.061 0.3013806 0.072 81 50 114 Stationary Solver 1 in Solver 1: Solution time: 0 s Physical memory: 471 MB Virtual memory: 569 MB
The solver also reports an error:
Failed to find a solution for all parameters, even when using the minimum parameter step. No convergence, even when using the minimum damping factor. Returned solution is not converged.
The beginning of the log file is as before, except that the solver now reports that the Parametric Solver is being called. We see that, for and , the solver completes. For , the solver fails and then automatically backtracks to try to find intermediate points that solve. Some of the intermediate steps are omitted for brevity, but we see that the parametric solver ends up very close to the analytic solution for the peak load. From this information, we could re-solve the problem with a different set of parameters and get a better idea of how the system behaves as we approach the failure load, which is often useful information.
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Consider the system shown below, of a spring that is attached to a rigid wall at one end, and with an applied force at the other end. The stiffness of the spring is a function of the distance it is stretched, . That is, the spring stiffness increases exponentially as it is stretched.
We are interested in finding the displacement of the end of the spring, where the force is applied. Just as we did earlier for the linear problem, we can now write the following function describing the balance of forces on the node for the nonlinear finite element problem:
In this case, only the spring stiffness is dependent on the solution, but more generally, both the load and the properties of the elements can be arbitrarily dependent upon the solution in a nonlinear problem.
Let us plot out this function, and keep in mind that we are trying to find such that .
Finding the solution to the problem is, in fact, only marginally different from the linear case. Recall that to solve the linear problem we took a single Newton-Raphson iteration — and we do the exact same thing here:
As you can see, we again start at an initial guess to the solution, , and evaluate the function, , as well as its derivative, . This gets us to the the point . By examination, we see that this is not the solution, since . But if we continue to take Newton-Raphson iterations, as shown below, it becomes clear that we are approaching the solution to the problem. (For more details about this algorithm, you can use this resource on Newton’s method.)
So finding the solution to a nonlinear problem is essentially identical to solving a linear problem, except that we take multiple Newton-Raphson steps to get to the solution. In fact, we could continue to take iterations and get arbitrarily close to the solution, but this is not needed. As discussed earlier, we always run into issues of numerical precision on computers, so there is a practical limit to how close we can get. Let’s have a look at the results after several iterations:
0 | 0.000 | 2.000 | ||
1 | 2.000 | 12.77 | 2.000 | 10.77 |
2 | 1.424 | 3.915 | 0.576 | 8.855 |
3 | 1.035 | 0.914 | 0.389 | 3.001 |
4 | 0.876 | 0.104 | 0.159 | 0.810 |
5 | 0.853 | 0.002 | 0.023 | 0.102 |
6 | 0.852 | 0.001 | 0.001 | 0.001 |
After six iterations, we see here that the difference between successive values of , and , as well as the absolute value of , is reduced to 0.001 or less. After six Newton-Raphson iterations starting from , the solution has converged to within a tolerance of 0.001. When we solve nonlinear problems, we apply this algorithm until the solution was converged to within the desired tolerance. There is a second termination criterion: that the solver should take no more than a specified number of iterations. Whichever criterion, tolerance, or number of iterations gets satisfied first will stop the solver. Also, keep in mind the discussion from the blog post on solving linear static finite element problems about the numerical scaling of the problem. The tolerance criteria applies to the scaled solution vector — not the absolute values of the solution.
Although it is more complicated to visualize, this is the same algorithm used to solve problems where is a vector, as is the case for typical nonlinear finite element problems. However, when solving a problem with hundreds, thousands, or even millions of degrees of freedom, it is desirable to take as few Newton-Raphson steps as possible. Recall that we need to solve and that computing the inverse of the derivative is the most computationally intensive step. To avoid proceeding into a region where there is no solution, and to minimize the number of Newton-Raphson steps taken, COMSOL uses a damping factor. Consider again the first Newton-Raphson step plotted earlier, and observe that for this step . So for this iteration, we have taken too large of a step. When this happens, COMSOL will perform a simple search along the interval for a point such that . The Newton-Raphson iteration scheme is then restarted at this point.
The term is known as the damping factor and has bounds . As we say that the damping is increased, while means that the problem is undamped. This method is attractive because the search requires only that COMSOL evaluates and the computational cost of this is quite low as compared to computing the derivative and its inverse .
It is important to emphasize that this damping term has no direct physical interpretation. Although this method works quite well to improve convergence, there is very little physical insight that can be gleaned by examining the damping factor. Furthermore, although COMSOL does allow you to manually modify the damping factor, it is not generally possible to use any physical understanding or information from the model as guidance when doing so. The default choice of damping algorithm is difficult to outperform through manual intervention. However, there are other techniques that can be used, which are usually motivated by the physics of the problem, that work well when the default damped Newton-Raphson methods converge slowly or not at all.
Nonlinear problems are inherently difficult to solve since there are multiple ways in which the above solution procedure can fail to converge. Although there are many ways in which the Newton-Raphson method can fail, in practice we can reduce the discussion to the following cases.
First, consider the same nonlinear problem as before, but with a different starting point, for example, . As we can see from the plot below, if we choose any initial condition , the Newton-Raphson method cannot find a solution since the derivatives of do not point towards the solution. There is no solution to be found to the left of , so these starting points are outside of the radius of convergence of the Newton-Raphson method. The choice of initial condition can cause the Newton-Raphson method to fail to converge, even if a solution exists. So, unlike the linear case, where a well-posed problem will always solve, the convergence of nonlinear models may be highly dependent on the choice of starting condition. We will address later how best to choose a good initial condition.
The nonlinear solver will also fail if the problem itself does not have a solution. Consider again the problem from above, but with a spring stiffness of . In other words, as the spring gets stretched, the stiffness decreases. If we plot out for a load of , we see that there is no solution to be found. Unfortunately, the Newton-Raphson algorithm cannot determine that this is the case; the algorithm will simply fail to find a solution and terminate after a user-specifiable number of iterations.
Last, consider the case of a material property that has a discontinuous change in properties. For example, consider the same system as before, but with a spring stiffness that has different values over different intervals, a value of for , a value of for , and for . If we plot out for this case we see that it is non-differentiable and discontinuous, which is a violation of the requirements of the Newton-Raphson method. It is also clear by examination that unless we choose a starting point in the interval the Newton-Raphson iterations will oscillate between iterations outside of this interval.
To summarize, so far we have introduced the damped Newton-Raphson method used to solve nonlinear finite element problems and discussed the convergence criteria used. We introduced several ways in which this method can fail to find a solution, including:
We will soon discuss ways of addressing all of these issues, but first, let’s take a look at the log file of a typical nonlinear finite element problem. Below you will see the log file (with line numbers added) from a geometric nonlinear structural mechanics problem:
1) Stationary Solver 1 in Solver 1 started at 10-Jul-2013 15:23:07. 2) Nonlinear solver 3) Number of degrees of freedom solved for: 2002. 4) Symmetric matrices found. 5) Scales for dependent variables: 6) Displacement field (Material) (mod1.u): 1 7) Iter ErrEst Damping Stepsize #Res #Jac #Sol 8) 1 6.1 0.1112155 7 3 1 3 9) 2 0.12 0.6051934 1.2 4 2 5 10) 3 0.045 1.0000000 0.18 5 3 7 11) 4 0.012 1.0000000 0.075 6 4 9 12) 5 0.0012 1.0000000 0.018 7 5 11 13) 6 1.6e-005 1.0000000 0.0015 8 6 13 14) Stationary Solver 1 in Solver 1: Solution time: 1 s 15) Physical memory: 849 MB 16) Virtual memory: 946 MB
Now you should have gained an understanding of how nonlinear static problems are solved in COMSOL as well as how to interpret the log file.
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Let’s consider a linear static finite element problem composed of three nodes and three elements:
Each element is bounded by two nodes. One of the nodes is at the rigid wall, where we know the displacement will be zero, so we do not need to solve for that node. As we saw in the earlier blog post on linear static finite element problems, we can write a balance of forces for each node:
and we can write this as:
or even more compactly as:
We can solve this problem using the Newton-Raphson iteration method, and since this is a linear static problem, we can solve it in one iteration and using an initial value of , giving us this solution:
Now, this problem only has two unknowns, or degrees of freedom (DOF), and can easily be solved with pen and paper. But in general, your matrices will have thousands to millions of DOF’s, and finding the solution to the above equation is usually the most computationally demanding part of the problem. When solving such a system of linear equations on a computer, one should also be aware of the concept of a condition number, a measure of how sensitive the solution is to a change in the load. Although COMSOL never directly computes the condition number (it is as expensive to do so as solving the problem) we do speak of the condition number in relative terms. This number comes into play with the numerical methods used to solve systems of linear equations.
There are two fundamental classes of algorithms that are used to solve for : direct and iterative methods. We will introduce both of these methods and look at their general properties and relative performance, below.
The direct solvers used by COMSOL are the MUMPS, PARDISO, and SPOOLES solvers. All of the solvers are based on LU decomposition.
These solvers will all arrive at the same answer for all well-conditioned finite element problems, which is their biggest advantage, and can even solve some quite ill-conditioned problems. From the point of view of the solution, it is irrelevant which one of the direct solvers you choose, as they will return the same solution. The direct solvers differ primarily in their relative speed. The MUMPS, PARDISO, and SPOOLES solvers can each take advantage of all of the processor cores on a single machine, but PARDISO tends to be the fastest and SPOOLES the slowest. SPOOLES also tends to use the least memory of all of the direct solvers. All of the direct solvers do require a lot of RAM, but MUMPS and PARDISO can store the solution out-of-core, which means that they can offload some of the problem onto the hard disk. The MUMPS solver also supports cluster computing, allowing you to use more memory than is typically available on any single machine.
If you are solving a problem that does not have a solution (such as a structural problem with loads, but without constraints) then the direct solvers will still attempt to solve the problem, but will return an error message that looks similar to:
Failed to find a solution. The relative residual (0.06) is greater than the relative tolerance. Returned solution is not converged.
If you get this type of error message, then you should check to make sure that your problem is correctly constrained.
The iterative solvers in COMSOL encompass a variety of approaches, but they are all conceptually quite simple to understand at their highest level, being essentially similar to a conjugate gradient method. Other variations include the generalized minimum residual method and the biconjugate gradient stabilized method, and there are many variations on these, but they all behave similarly.
Contrary to direct solvers, iterative methods approach the solution gradually, rather than in one large computational step. Therefore, when solving a problem with an iterative method, you can observe the error estimate in the solution decrease with the number of iterations. For well-conditioned problems, this convergence should be quite monotonic. If you are working on problems that are not as well-conditioned, then the convergence will be slower. Oscillatory behavior of an iterative solver is often an indication that the problem is not properly set up, such as when the problem is not sufficiently constrained. A typical convergence graph for an iterative solver is shown below:
By default, the model is considered converged when the estimated error in the iterative solver is below 10^{-3}. This is controlled in the Solver Settings window:
This tolerance can be made looser, for faster solutions, or tighter, for greater accuracy on the current mesh. The tolerance must always be greater than a number that depends on the machine precision (2.22×10^{-16}) and the condition number (which is problem dependent). However there is usually no point in making the tolerance too tight since the inputs to your model, such as material properties, are often not accurate to more than a couple of digits. If you are going to change the relative tolerance, we generally recommend making the tolerance tighter in increments of one order of magnitude and comparing solutions. Keep in mind that you are only solving to a tighter tolerance on the mesh that you are currently using, and it is often more reasonable to refine the mesh.
The big advantage of the iterative solvers is their memory usage, which is significantly less than a direct solver for the same sized problems. The big disadvantage of the iterative solvers is that they do not always “just work”. Different physics do require different iterative solver settings, depending on the nature of the governing equation being solved.
Luckily, COMSOL already has built-in default solver settings for all predefined physics interfaces. COMSOL will automatically detect the physics being solved as well as the problem size, and choose the solver — direct or iterative — for that case. The default iterative solvers are chosen for the highest degree of robustness and lowest memory usage, and do not require any interactions from the user to set them up.
When solving the systems of linear equations of a simulation, COMSOL will automatically detect the best solver without requiring any user interaction. The direct solvers will use more memory than the iterative solvers, but can be more robust. Iterative solvers approach the solution gradually, and it is possible to change the convergence tolerance, if desired.
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As we saw earlier, there are four different 3D element types — tets, bricks, prisms, and pyramids:
These four elements can be used, in various combinations, to mesh any 3D model. (For 2D models, you have triangular and quadrilateral elements available. We won’t discuss 2D very much here, since it is a logical subset of 3D that doesn’t require much extra explanation.) What we haven’t spoken in-depth about yet is why you would want to use these various elements.
Tetrahedral elements are the default element type for most physics within COMSOL Multiphysics. Tetrahedra are also known as a simplex, which simply means that any 3D volume, regardless of shape or topology, can be meshed with tets. They are also the only kind of elements that can be used with adaptive mesh refinement. For these reasons, tets can usually be your first choice.
The other three element types (bricks, prisms, and pyramids) should be used only when it is motivated to do so. It is first worth noting that these elements will not always be able to mesh a particular geometry. The meshing algorithm usually requires some more user input to create such a mesh, so before going through this effort, you need to ask yourself if it is motivated. Here, we will talk about the motivations behind using brick and prism elements. The pyramids are only used when creating a transition in the mesh between bricks and tets.
It is worth giving a bit of historical context. The mathematics behind the finite element method was developed well before the first electronic computers. The first computers to run finite element programs were full of vacuum tubes and hand-wired circuitry, and although the invention of transistors led to huge improvements, even the supercomputers from 25 years ago had about the same clock speed as today’s fashion accessories.
Some of the first finite element problems solved were in the area of structural mechanics, and the early programs were written for computers with very little memory. Thus, first-order elements (often with special integration schemes) were used to save memory and clock cycles. However, first-order tetrahedral elements have significant issues for structural mechanics problems, whereas first-order bricks can give accurate results.
As a legacy of these older codes, many structural engineers will still prefer bricks over tets. In fact, the second order tetrahedral element used for structural mechanics problems in the COMSOL software will give accurate results, albeit with different memory requirements and solution times from brick elements.
The primary motivation in COMSOL Multiphysics for using brick and prism elements is that they can significantly reduce the number of elements in the mesh. These elements can have very high aspect ratios (the ratio of longest to shortest edge), whereas the algorithm used to create a tet mesh will try to keep the aspect ratio close to unity. It is reasonable to use high aspect ratio brick and prism elements when you know that the solution varies gradually in certain directions or if you are not very interested in accurate results in those regions because you already know the interesting results are elsewhere in the model.
Consider the example of a wheel rim, shown below.
The mesh on the left is composed only of tets, while the mesh on the right has tets (green), bricks (blue), and prisms (pink), as well as pyramids to transition between these elements. The mixed mesh uses smaller tets around the holes and corners, where we expect higher stresses. Bricks and prisms are used in the spokes and around the rim. Neither the rim nor the spokes will carry peak stresses (at least under a static load), and we can safely assume a relatively slow variation of the stresses in these regions.
The tet mesh has about 145,000 elements and around 730,000 degrees of freedom. The mixed mesh has close to 78,000 elements and roughly 414,000 degrees of freedom, taking about half as much time and memory to solve. The mixed mesh does take significant user interaction to set up, while the tet mesh requires essentially no user effort.
Note that there is not a direct relationship between degrees of freedom and memory used to solve the problem. This is because the different element types have different computational requirements. A second-order tet has 10 nodes per element, while a second-order brick has 27. This means that the individual element matrices are larger, and the corresponding system matrices will be denser, when using a brick mesh. The memory (and time) needed to compute a solution depends upon the number of degrees of freedom solved for, as well as the average connectivity of the nodes, and other factors.
Another example is shown below. This time, it’s a structural analysis of a loaded spring. Since the deformation is quite uniform along the length of the helix of the spring, it makes sense to have a mesh that describes the overall shape and cross section, but relatively stretched elements along the length of the wire. The prism mesh has 504 elements with 9526 degrees of freedom, and the tet mesh has 3652 elements with 23,434 degrees of freedom. So although the number of elements is quite different, the number of degrees of freedom is less so.
The other significant motivation for using brick and prism elements is when the geometry contains very thin structures in one direction, such as an epitaxial layer of material on a wafer, a stamped sheet metal part, or a sandwiched composite.
For example, let’s look at the figure below of a thin trace of material patterned onto a substrate. The tet mesh has very small elements in the trace, whereas the prism mesh is composed of thin elements in this region. Whenever your geometry has layers that are about 10^{-3} or so times thinner than the largest dimension of the part, the usage of bricks and prisms becomes very highly motivated.
It is also worth pointing out that the COMSOL software offers many boundary conditions that can be used in lieu of explicitly modeling thin layers of materials. For example, in electromagnetics, the following four tutorial models consider thin layers of material with relatively high and low conductivity, and relatively high and low permeability:
Similar types of boundary conditions exist in most of the physics interfaces. Usage of these types of boundary conditions will avoid the need to mesh such thin layers entirely.
Lastly, the above comments apply only to linear static finite element problems. Different meshing techniques are needed for nonlinear static problems, or if we are modeling time-domain or frequency-domain phenomena.
To summarize, here is what you should keep in mind when starting your meshing of linear static problems:
Let’s look at the problem of a flat plate under uniaxial tension with a square hole cut in it. This is similar to the example from earlier in that we can exploit symmetry and only model one quarter of the structure.
As before, we can use adaptive mesh refinement to let COMSOL insert more elements into regions where the error is estimated to be large:
We observe that smaller and smaller elements are being inserted at the inside corner of the sharp hole. Let’s also plot out the stresses at this inside corner as a function of the mesh size:
From this plot, it appears that the stresses are getting larger and larger, no matter how fine we make the mesh. In fact, that is exactly what is occurring here; the stresses at the sharp corner are non-convergent with respect to mesh refinement because we have a singularity in the model. This is actually completely accurate — stresses in sharp corners are theoretically infinite. Whenever you see this kind of non-convergent behavior, it is likely that you are looking at the manifestation of a singularity in your model.
In structural engineering practice, sharp inside corners are something to avoid. You would be justified in saying that one way of preventing this problem is to round off the sharp corners of the model where these singularities appear. Doing so would lead to a model that predicts stresses that converge with mesh refinement, but it would still need to have a lot of elements in this inside corner. So let’s introduce other approaches to dealing with these singularities.
One approach to dealing with these singularities is to just ignore them. An important feature to understand about the finite element method is that it allows local inaccuracies, since it is formulated in a way that minimizes the global error in the model. The stresses we are predicting in the above model are incorrect, but if you evaluate the stresses at a distance of around 2-3 mesh elements away from the singularity, the stress solution there does converge. Thereby, if we’re interested in the stresses away from the singularity, the mere presence of the singularity does not pollute the predictions elsewhere.
It is also important to realize that the singularities manifest when you are taking the derivative of the solution field. In structural mechanics, we solve for the displacement field, , and compute the stresses from the strains, , where strain is defined in terms of the gradient of the displacement field: . When you think of the stress as being the gradient of the displacement field, it also becomes a little bit clearer why the solution for the stresses goes to infinity at a sharp corner. However, if you are only interested in the solution field, , this is not singular, even at the sharp corner, and does converge with mesh refinement.
There is one more common case when a singularity is acceptable: if you are only interested in an integral quantity as output of your model. For instance, the total elastic strain energy of a system composed of a linear material is:
If we evaluate this for a domain that includes a singularity, such as the plate with the square hole, this integral will converge rapidly with mesh refinement, even though the integrands are non-convergent at point(s) inside the domain. So if the only quantity that you want to get out is a function of an integral over a domain (or boundary) within your model, then your model can include singularities. The quantity that you are integrating will converge to the same value if you use a sharp or a round corner. This situation arises especially often in electromagnetics, where device inductance and capacitance are both evaluated as integrals of electric and magnetic fields over the domains.
To summarize, there are three common situations when it is acceptable to have a singularity in the model:
In these cases, you will observe convergence of your solution with mesh refinement. That said, you should still be careful when you observe non-convergent behavior anywhere in your model, and make sure that it is not skewing your interpretation of the results.
Finally, there are cases where we do need to accurately compute the fields at these singularities, but our model may be so large that we don’t want to put fillets on all of these edges. In such cases, we can use a strategy called submodeling, or break-out modeling. This approach uses a relatively coarse mesh to find the solution field on a larger model that may contain singularities, and then passes this information on to a submodel that has a much finer mesh and rounded corners. This approach is presented in the Submodel in a Wheel Rim example.
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The finite element mesh serves two purposes. It first subdivides the CAD geometry being modeled into smaller pieces, or elements, over which it is possible to write a set of equations describing the solution to the governing equation. The mesh is also used to represent the solution field to the physics being solved. There is error associated with both the discretization of the geometry as well as discretization of the solution, so let’s examine these separately.
Consider two very simple geometries, a block and a cylindrical shell:
There are four different types of elements that can be used to mesh these geometries — tetrahedra (tets), hexahedra (bricks), triangular prismatics (prisms), and pyramid elements:
The grey circles represent the corners, or nodes, of the elements. Any combination of the above four elements can be used. (For 2D modeling, triangular and quadrilateral elements are available.) You can see by examination that both of these geometries could be meshed with as few as one brick element, two prisms, three pyramids, or five tets. As we learned in the previous blog post about solving linear static finite element problems, you will always arrive at a solution in one Newton-Raphson iteration. This is true for linear finite element problems regardless of the mesh. So let’s take a look at the simplest mesh we could put on these structures. Here’s a plot of a single brick element discretizing these geometries:
The mesh of the block is obviously a perfect representation of the true geometry, while the mesh of the cylindrical shell appears quite poor. In fact, it only appears that way when plotted. Elements are always plotted on the screen as having straight edges (this is done for graphics performance purposes) but COMSOL usually uses a second-order Lagrangian element to discretize the geometry (and the solution). So although the element edges always appear straight, they are internally represented as:
The white circles represent the midpoint nodes of these second-order element edges. That is, the lines defining the edges of the elements are represented by three points, and the edges approximated via a polynomial fit. There are also additional nodes at the center of each of these quadrilateral faces and in the center of the volume for these second-order Lagrangian hexahedral elements (omitted for clarity). Clearly, these elements do a better job of representing the curved boundaries of the elements. By default, COMSOL uses second-order elements for most physics, the two exceptions are problems involving chemical species transport and when solving for a fluid flow field. (Since those types of problems are convection dominated, the governing equations are better solved with first-order elements.) Higher order elements are also available, but the default second-order elements usually represent a good compromise between accuracy and computational requirements.
The figure below shows the geometric discretization error when meshing a 90° arc in terms of the number of first- and second-order elements:
The conclusion that can be made from this is that at least two second-order elements, or at least eight first-order elements, are needed to reduce the geometric discretization error below 1%. In fact, two second-order elements introduce a geometric discretization error of less that 0.1%. Finer meshes will more accurately represent the geometry, but will take more computational resources. This gives us a couple of good practical guidelines:
With these rules of thumb, we can now estimate the error we’ve introduced by meshing the geometry, and we can do so with some confidence before even having to solve the model. Now let’s turn our attention to how the mesh discretizes the solution.
The finite element mesh is also used to represent the solution field. The solution is computed at the node points, and a polynomial basis is used to interpolate this solution throughout the element to recover the total solution field. When solving linear finite elements problems, we are always able to compute a solution, no matter how coarse the mesh, but it may not be very accurate. To understand how mesh density affects solution accuracy, let’s look at a simple heat transfer problem on our previous geometries:
A temperature difference is applied to opposing faces of the block and the cylindrical shell. The thermal conductivity is constant, and all other surfaces are thermally insulated.
The solution for the case of the square block is that the temperature field varies linearly throughout the block. So for this model, a single, first-order, hexahedral element would actually be sufficient to compute the true solution. Of course, you will rarely be that lucky!
Therefore, let’s look at the slightly more challenging case. We’ve already seen that the cylindrical shell model will have geometric discretization error due to the curved edges, so we would start this model with at least two second-order (or eight first-order) elements along the curved edges. If you look closely at the above plot, you can see that the element edges on the boundaries are curved, while the interior elements have straight edges.
Along the axis of the cylinder, we can use a single element, since the temperature field will not vary in this direction. However, in the radial direction, from the inside to outside surface, we also need to have enough elements to discretize the solution. The analytic solution for this case goes as and can be compared against our finite element solution. Since the polynomial basis functions cannot perfectly describe the function, let’s plot the error in the finite element solution for both the linear and quadratic elements:
What you can see from this plot is that, as you increase the number of elements in the model, the error goes down. This is a fundamental property of the finite element method: the more elements, the more accurate your solution. Of course, there is also a cost associated with this. More computational resources, both time and hardware, are required to solve larger models. Now, you’ll notice that there are no units to the x-axis of this graph, and that is on purpose. The rate at which error decreases with respect to mesh refinement will be different for every model, and depends on many factors. The only important point is that it will always go down, monotonically, for well-posed problems.
You’ll also notice that, after a point, the error starts to go back up. This will happen once the individual mesh elements start to get very small, and we run into the limits of numerical precision. That is, the numbers in our model are smaller than can be accurately represented on a computer. This is an inherent problem with all computational methods, not just the finite element method; computers cannot represent all real numbers accurately. The point at which the error starts to go back up will be around and to be on the safe and practical side, we often say that the minimal achievable error is 10^{-6}. Thus, if we integrate the scaled difference between the true and computed solution over the entire model:
We say that the error, , can typically be made as small as 10^{-6} in the limits of mesh refinement. In practice, the inputs to our models will anyways usually have much greater uncertainty than this. Also keep in mind that in general we don’t know the true solution, we will instead have to compare the computed solutions between different sized meshes and observe what values the solution converges toward.
I would like to close this blog post by introducing a better way to refine the mesh. The plots above show that error decreases as all of the elements in the model are made smaller. However, ideally you would only make the elements smaller in regions where the error is high. COMSOL addresses this via Adaptive Mesh Refinement, which first solves on an initial mesh and iteratively inserts elements into regions where the error is estimated to be high, and then re-solves the model. This can be continued for as many iterations as desired. This functionality works with triangular elements in 2D and tetrahedrals in 3D. Let’s examine this problem in the context of a simple structural mechanics problem — a plate under uniaxial tension with a hole, as shown in the figure below. Using symmetry, only one quarter of the model needs to be solved.
The computed displacement fields, and the resultant stresses, are quite uniform some distance away from the hole, but vary strongly nearby. The figure below shows an initial mesh, as well as the results of several adaptive mesh refinement iterations, along with the computed stress field.
Note how COMSOL preferentially inserts smaller elements around the hole. This should not be a surprise, since we already know there will be higher stresses around the hole. In practice, it is recommended to use a combination of adaptive mesh refinement, engineering judgment, and experience to find an acceptable mesh.
Consider the system shown below, of a spring that is attached to a rigid wall at one end, and with an applied force at the other end.
We are interested in finding the displacement of the end of the spring, where the force is applied. Using the vocabulary of the finite element method, we have here a single element finite element model. The spring is the element, and it is bounded by two nodes at either end. One of the nodes is fixed rigidly to the wall, and other node will deform due to the applied load. We are trying to find the nodal displacements due to the applied load. This problem is linear because neither the material properties (the spring constant) nor the loads are dependent upon the solution. This is a static problem because we are finding the solution under the assumption that there is no variation with respect to time.
This problem can easily be solved with pen and paper, but let’s look at how this is done in a more rigorous way. Consider the node where we are trying to find the displacement, and draw the balance of forces at equilibrium:
We can write this out as: , and we call this equation the functional of the system, at equilibrium (steady state conditions) the functional is equal to zero. We want to find the value of such that . Let’s also plot out the functional:
Of course we can find the solution by examination in this case, but in general will be a vector with possibly millions of unknowns, so let’s be more rigorous and take a look at the exact algorithm used:
The above algorithm is also known as the Newton-Raphson method. Additionally, we can visualize this graphically as:
Note that regardless of the starting point, the solution will be found in one step. So, whenever you solve a linear static finite element problem in COMSOL Multiphysics, the software is following this algorithm to find the solution. Now, the above example has only a single unknown, so we only need to solve a single linear equation. Typically your models will have thousands or even millions of unknowns, which means you will need to solve a system of linear equations, but the idea is the same.
Lastly, we address the issue of numerical scaling. Whenever we solve a problem on a computer, we need to consider the problem of finite precision due to the floating point representation of numbers. Computers cannot represent the space of the real numbers perfectly. To minimize the effect of this, COMSOL applies a scale factor to the equations before they are solved. COMSOL automatically chooses a scaling appropriate for each set-up field variables in the model, and this almost never needs to be handled by the user, but it is worth knowing what effect the scale factor has. As long as it is within a few orders of magnitude of the average magnitude of the values in the solution, then no interaction is required. Only if the scaling is very different from the expected solution magnitude should the scale factor be changed.
To summarize what you need to know about solving linear static finite element models:
Now let’s look at how the above information can be used to interpret the COMSOL Log file of a typical linear static finite element model. Here’s the log file (with line numbers added) from a thermal stress problem:
1) Stationary Solver 1 in Solver 1 started at 30-Apr-2013 17:41:45. 2) Linear solver 3) Number of degrees of freedom solved for: 3651 (plus 124 internal DOFs). 4) Symmetric matrices found. 5) Scales for dependent variables: 6) Displacement field (Material) (mod1.u): 0.0090 7) Temperature (mod1.T): 2.9e+002 8) Iter Damping Stepsize #Res #Jac #Sol 9) 1 1.0000000 4.7e-017 1 1 1 10) Stationary Solver 1 in Solver 1: Solution time: 0 s 11) Physical memory: 878 MB 12) Virtual memory: 879 MB
Now you should have an understanding of how linear static problems are solved in COMSOL, and how to interpret the log file.
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