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Solver Sequences and Settings
The Study Node
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Study Step Study Step
Controls: • Which Meshes to use • Which Physics Interfaces to solve for • Some fundamental settings for the study type (e.g. the continuation solver for a Stationary or the time span for a Time Dependent)
Solver Sequence
Solver Sequence
Controls the sequence of: • Study Steps • Variables (unknowns) used • Solvers used
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Study Step Pointer
Pointer to Study Step
Note: May point to Study Step of different Study: Study 2, Study 3 etc.
Variables • Controls: • Variables Not Solved For • Initial Values • Scaling • Corresponds roughly to Old Solver Manager Init Page
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Solver Settings
Corresponds to Solver Parameters of 3.5a. Solver tolerances are set here
Computing a Solution Automatically generate a Solver Sequence corresponding to the Study steps and compute the solution.
Only generate a Solver Sequence corresponding to the Study Steps. For manual inspection/editing. No computations.
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Computing a Solution: When Editing Low-Level Settings Compute Using Attached Sequence Compute Using Attached Sequ ence will use
a Solver Sequence that you have manually edited.
Here: using Compute instead would generate a new default Solver Sequence to avoid overwriting your changes
Computing a Solution: Sequence Level Use this Solver Sequence
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Computing a solution Use the nodes up to and including this node (here Variables).
Use all nodes in the Solver Sequence
Study Type: Bundle of Several Study Steps
One Study Type (Frequency Domain Model) = Two Study Steps (Eigenfrequency & Frequence Domain Modal)
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The Study Node • Study defines how to solve a model – Study Steps – Solver Configurations – Job Configurations
Check Show More Options to see: • Solver Configurations • Job Configurations
Study Steps •
You can add multiple study steps by right -clicking on the Study node
•
COMSOL automatically uses the solution of the first study step as the initial value for the second study step
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Solver Configurations •
Contains all the solver settings and sequences
•
This is where you “tune” the solver
•
To look at the Solver Configurations before solving, right -click on Solver Configurations and select Show Default Solver
Solver Configurations – More details
Information on study step
Information on variables being solved and their initial values
Detailed information on solver settings such as choice of solver, tolerance, etc.
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Stationary Solver Settings
Stationary solver settings …Solver Sequence 1 > Stationary 1
Stop condition for Newton iterations
• COMSOL can inspect the stiffness matrix and decide whether the problem is linear or nonlinear • We can also explicitly choose linear or nonlinear
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Solver settings – more information …Solver Sequence 1 > Stationary 1 > Fully Coupled 1
Linear system of equations is solved using the specified solver
Can choose different damping methods
Stop condition for Newton iterations
Revisit: k = (0.1+10*(T>25[K])) Instead of using a “jump” in k, use a smoothing function Now try: k = 0.1+DK*flc2hs(T-25,STEP) Try: DK, STEP = 0 8 2.5 4 5 2 7.5 0.5 10 0.25
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Achieving convergence for non-linear problems DONE
Assuming that the problem is well-posed, try solving it
Converging?
Not converging?
1) Check the initial condition Not converging? Converging?
2) Use parametric solver to ramp load Not converging?
Perform a mesh refinement study. Try a finer mesh and check that the solution is similar.
3) Use parametric solver to ramp non-l inearity Not converging?
Not converging?
4) Refine the mesh Not converging?
Does the problem have a steady-state solution? Are there additional effects? Check all assumptions.
Solving the Linear System Matrix •
Linear system matrix
•
Direct methods
•
Iterative methods
•
Suggestions on solver selection
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Let’s take another look at the system equations 0 k k
p k 2
f u
k u 0 b Ku 0 k k u 0
3
3
3
1
2
3
3
Define a quadratic function: r (u) = b·u-½Ku·u r (u)
u3
The solution, f (usolution) = 0, is the point where r (u), is at a minimum
Solution
u2
Finding the minimum of a quadratic function r (u) = 2u2 - 3u + 1 Newton’s method: u so l u 0
r (u)
r u 0 r u 0
For a quadratic function, this converges in one iteration, for any u0 usol u
r (u) = b·u-½Ku·u r′ (u) = b - Ku r ″ (u) = - K r u 0
r′ (u) =
4u - 3 r ″ (u) = 4 u0 = 0 u sol u0
r u0 r u0
0
Via our choice of r (u), this reduces to the same equation from the previous section:
u sol
40 3 4
0.75
u0
u0 u sol
0 0
r u 0 b
K
u0
b Ku 0
K
K 1b
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Finding the minimum of the quadratic function, r (u), by the direct method means solving u=K -1b •
This is known as Gaussian Elimination, or LU factorization
•
The numerical algorithms are beyond the scope of this course, but they have the following important properties: – For 3D, requires O(n2)-O(n3) numeric operations, where n is the length of u – Requires O(n2)-O(n3) memory – Robustness of the algorithm is onl y very weakly dependent upon K
•
The direct solvers in COMSOL are: – MUMPS: fast, required for cluster computation – PARDISO: fast, multi-core capable – SPOOLES: slow, uses the least memory
Introduction to iterative methods for finding the minimum of a quadratic function, a naive approach 1) Start here 2) Search along coordinate axis 3) Find the minimum along that axis 4) Repeat until converged
This iterative method requires only that we can repeatedly evaluate r (u) as opposed to the direct methods, which get to the minimum in one step by evaluating [r″ (u)]-1
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Ill-conditioned matrices require more iterations
Numerical error becomes significant
A better iterative method for finding the minimum: The Conjugate Gradient (CG) method 1) Start here 2) Initially, find the gradient vector 3) Find the minimum along that vector 4) Find the conjugate gradient vector * 5) Repeat 3-4 until converged The CG method requires that we can evaluate r (u), r ′( u) and r ″ (u) CG converges in at most n iterations
CG does NOT compute K-1
*) See e.g. Wikipedia, or “Scientific Computing” by Michael Heath
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Condition number matters for iterative methods
A matrix with a condition number of 1 will converge in one iteration, (rare!)
Numerical error becomes significant for ill-conditioned matrices
All iterative methods in COMSOL are some variation upon the CG method Conjugate Gradient, GMRES, FGMRES and BiCGStab The details of these algorithms are beyond the scope of this cou rse
All these methods make use of PRECONDITIONERS The system equation, Ku = b, is multiplied by a precondi tioner matrix, M, to improve the condition number
Exercise: Show that the best possible preconditioner is the matrix M=K-1
Ku = b
MKu = Mb
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What you need to know about iterative solvers • •
They converge in at most n iterations (good) Solution time is O(n1-n2) (good) – Solution time depends on condition number
• •
Memory requirements are O(n1-n2) (very good) They are less robust that the direct solvers (neutral) – Convergence depends upon condition number – An ill-conditioned problem is often set up incorrectly
•
Different physics require different iterative methods (bad) – – – –
This is an ongoing research topic Often cannot solve two physics with the sam e solver Improvements in methods are ongoing We have tried to find the best combination of iterative solvers and preconditioners for many physics, to find these settings, read the manuals or open a new model fi le, select a space dimension of 3 D and the physics you want to solve – GMRES and incomplete LU, is usually the best choice for multiphysics problems
Choosing the linear system solver
• COMSOL chooses the optimized solver and its settings based on the chosen space dimension, physics and study type • We can also choose a different solver - Not recommended in general - Could be useful for multiphysics problems
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Solver selection for 1D and 2D problems •
Begin with the default solver, MUMPS, PARDISO or SPOOLES
•
If running out of memory: – Use MUMPS or PARDISO out-of-core (write LU factors on hard drive, S L O W ) – Upgrade your computer, 64-bit & more memory – Make sure that your problem is not overly complicated, typical 2D models should solve easily on a 32-bit computer with 2GB of RAM
•
If you are seeing differences between the solvers, check the problem very carefully, this usually indicates a m istake in the model – For some nonlinear problems (e.g. fluid-flow), set “Check Error Estimate” as “No” in Study 1 > Solver Sequences > Solver Sequence 1 > C h o s e n s t u d y > Direct
Solver selection for 3D problems •
Set up a linear problem first
•
Use the default solver settings for the physics you are solving
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If you run out of memory, upgrade to 64-bit and more RAM
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Monitor the memory requirements as you grow the problem size
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Setup a non-linear problem only after you have successfully solved a linear problem
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Important notes on solvers •
Direct – Robustness of the algorithm is o nly very weakly dependent upon K – The direct solvers in COMSOL are: • PARDISO: fast, BEST FIRST CHOICE FOR SHARED MEMORY • SPOOLES: slow, uses the least memory (Can run distributed) • MUMPS: slow, robust, uses the most memory BEST FIRST CHOICE FOR CLUSTER
•
Iterative – Slower than Direct – Convergence depends on condition number of K – Different physics require different solver configurations (bad) • This is an ongoing research topic • Often cannot solve two physics with the same solver • Improvements in methods are ongoing • We have tried to find the best combination of iterative solvers and preconditioners for many physics, to find these settings, read the manuals or open a new model file, select a space dimension of 3D and the physics you want to solve
Time Dependent Solver Settings
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Time-dependent problems •
Time-dependent formulation
•
Tolerance and other attributes of time-dependent solver
•
Suggestions on convergence of time-dependent problems
Time-dependent Study
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COMSOL picks its own optimum timesteps These are the output timesteps, the times you want to see the solution at. They do not affect the internal timesteps. The internal timesteps are controlled by the . TOLERANCES
The absolute tolerance can be different for each solution variable Temperature: Absolute tolerance: ±0.1K
Voltage: Absolute tolerance: ±0.001V
Often need to set this when solving multi-physics problems
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Absolute and Relative Tolerance u
True solution t
u
Solutions within Relative tolerance
At solutions near zero, the relative tolerance cannot be used, since it would result in very small timesteps
t
u
Solutions within T i g h t en b o t h t h e Absolute tolerance absolu te and relative t o l e r an c e u n t i l y o u t are satisfied w ith the results ov er time.
u t
Other timestepping options Selecting the physics at startup will usually load the appropriate settings. For wave type problems, Generalized-alpha is often better. Free - time stepping method chooses time steps freely Intermediate - force the time stepping method to take at least one step in each subinterval of the times specified
BDF order, set max and min to 2 and 1 for wave problems.
Strict - force the time stepping method to take steps that end at the times specified
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More timestepping options Specify for which steps the solutions should be stored
Give a condition to stop at, when this expression becomes negative
Make sure not to impose any “ringing” loads, unless they are really there Load
time Response
time
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A reasonable starting point is necessary • • • •
The load applied at t = 0 must agree with the initial condition It is helpful to a have a load with time derivative zero at t = 0 The Physics > Initial Values node is used to set initial conditions The Solver Configurations > Solver > Dependent Variables > Initial Values area can be used to overwrite that information Most common mistake:
b
Initial conditions for fluid flow: u = 0 Velocity/Pressure inlet condition that does NOT ramp up from zero t u
u(tinitial)
= [ K steady-state]
-1
b(tinitial)
Solution: Use flc1hs function to ramp load and to keep d b /dt (t=0)=0 t
Segregated Solvers
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Segregation •
Coupled solver
•
Segregated solver
Physics 1
Physics 2
Physics 3
Physics 1
Physics 2
Physics 3
Segregation •
Easier to obtain good starting guesses for strongly nonlinear Multiphysics models and time-dependent models
•
Extremely memory efficient
•
Different solver settings for each step – Good for iterative choices – Solver – Damping
Microwave-thermal-structural Multiphysics couplings
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Using the segregated solver Study > Solver Sequences > Solver Sequence > Stationary
Can add multiple Segregated steps as required
Segregated step Study > Solver Sequences > Solver Sequence > Stationary > Segregated
Choose the variable/s that you want to solve in this segregated step Choose the linear system solver and the damping method for each segregated step
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Recap: Working with solvers • A problem could be Linear or Nonlinear •
The linear system of equations could be solved using a Direct Solver or an Iterative Solver
• A multiphysics problem could be solved using a Fully-Coupled Approach or Segregated Approach
Solver Sequences and Settings
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