MMS for 2-D Steady Heat Conduction — Troubleshooting Guide
By Sitemap
My Mms Heat 2d simulation is giving me unexpected results — convergence issues, maybe. How do I diagnose this systematically?
Mms Heat 2d troubleshooting follows patterns once you know what to look for. Most issues fall into three buckets: convergence failures, accuracy problems, and result misinterpretation. Let me give you a systematic diagnostic framework rather than a list of random fixes.
That framing helps. Before we dive in — what's the single most common mistake engineers make with Mms Heat 2d?
Honestly, it's skipping the sanity checks. Engineers set up a Mms Heat 2d model, it converges, and they trust the result without verifying it against a hand calculation or a known benchmark. The solver gives you an answer regardless of whether your model is physically correct. Always run a simplified version first.
A steady heat-conduction MMS is the ideal entry point to code verification, because the source term can be worked out by hand. Take the unit square with a constant conductivity \( k \) and pick the manufactured solution \( T_m = \sin(\pi x)\sin(\pi y) \), and you get
$$ s = -k\,\nabla^2 T_m = 2\pi^2 k \,\sin(\pi x)\sin(\pi y) $$
in a single line. The boundary can be \( T_m = 0 \) (Dirichlet) all the way around, so no symbolic computation is needed at all. The shortest learning path is to absorb the shape of a verification study — mesh sequence, error norm, observed order — on this one fully checkable problem, and only then move on to elasticity and fluids. Even so, this simple problem has several places to get stuck; this page clears them symptom by symptom.
| Check | What to look for |
|---|---|
| Sign of the source term | The sign convention follows how the equation is written (\( -\nabla\cdot(k\nabla T) = s \) versus \( \nabla\cdot(k\nabla T) + s = 0 \)). The wrong sign flips the solution upside down — one glance at the temperature contours settles it |
| Forgotten coefficient | Drop the \( k \) from \( 2\pi^2 k \) and the shape of the solution is right while the amplitude is off by a factor of \( 1/k \). "Right shape, wrong amplitude" is this pattern |
| Unassigned boundary | In FEM an unspecified edge is automatically insulated (zero Neumann). Missing the assignment on even one edge changes the solution substantially. Verify the boundary assignments visually |
| Unit system | Feeding W/m³ into a millimetre-based model is off by a factor of 10⁹. Compare against a hand-calculated order of magnitude |
The natural next step is to introduce \( k(T) \), and the easy thing to overlook is that the source-term derivation changes. Correctly,
$$ s = -\nabla\cdot\left(k(T_m)\,\nabla T_m\right) = -k(T_m)\nabla^2 T_m - k'(T_m)\,|\nabla T_m|^2 $$
and forgetting the second term (the \( k' \) term) is the standard mistake. From here on it is safer to switch to symbolic derivation (SymPy or similar). The problem is also nonlinear now, so drive the iteration (Picard/Newton) two decades below the discretization error — a shallowly converged nonlinear run will happily destroy the order you were getting in the linear case.
This simple problem is usually where you first meet the "error floor". The causes are ① the convergence criterion of the linear solver (for iterative methods), ② round-off (once the error reaches the \( 10^{-10} \) range), and ③ approximations mixed into the evaluation of the source term or boundary values (a coarse interpolation table, for instance). Diagnosis takes two moves: look at the absolute value of the error, and tighten the convergence criterion by one decade to see what changes. Going all the way to "refine until you hit the floor, identify what the floor is made of, and report it" is what raises a verification report from adequate to complete.
Whether MMS is feasible depends less on the solver itself than on "can you define a coordinate-dependent volumetric heat source?" The routes in the major tools: Ansys Mechanical = function definition (Function, or %table% and coordinate functions in APDL); Abaqus = an Analytical Field or a DFLUX subroutine; COMSOL = type the expression directly (the easiest); OpenFOAM = codedSource/fvOptions; Fluent = a UDF (DEFINE_SOURCE). Even in tools where no expression can be entered from the GUI, a fine spatial table plus interpolation serves as a substitute (make it fine enough that the interpolation error does not pollute the order — see Symptom 2). Establish this "MMS route in my own tool" once, and every verification study afterwards reuses it.
Honestly, I used to wonder whether a problem this easy was worth the trouble…
The real deliverable here is not "I solved it" but the experience of running the whole verification procedure end to end, and a reusable set of scripts. Building the mesh sequence, computing the error norm, plotting the order, recognizing the floor — the tools you build here carry over unchanged to elasticity and to NS. And because the problem is simple, bugs in your procedure never get tangled up with bugs in the physics. It is like practising scales: you build the form on an easy exercise so that the hard pieces become playable. As a one-day investment, it is the highest-yield single problem in code verification.
Related: MMS overview (consolidated), convergence-order troubleshooting, 2-D elasticity MMS (the next step).