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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 Sequence 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

Stop condition for

Newton iterations

Can choose different

damping methods

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

Assuming that the problem is

well-posed, try solving it

1) Check the initial condition

2) Use parametric solver to ramp load

3) Use parametric solver to ramp non-linearity

4) Refine the mesh

Does the problem have a steady-state solution?

Are there additional effects? Check all assumptions.

Perform a mesh

refinement study.

Try a finer mesh

and check that the

solution is similar.

DONE

Not converging?

Not converging?

Not converging?

Not converging?

Not converging?

Converging?

Not converging?

Converging?

Solving the Linear System Matrix

Linear system matrix

Direct methods

Iterative methods

Suggestions on solver selection

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Lets take another look at the system equations

0Kubuf

0

00

3

2

313

332

u

u

kkk

kkk

p

Define a quadratic function: r(u) = bu-Kuu

u2

u3 r(u)

Solution The solution, f(usolution) = 0, is

the point where r(u), is at a

minimum

Finding the minimum of a quadratic function

r (u) = 2u2 - 3u + 1

r (u)

u

Newtons method: 0

00

u

uuu

r

rsol

For a quadratic function, this

converges in one iteration, for any u0

r(u) = 4u - 3

r(u) = 4

75.04

3040

0

00

ur

uruusol

u0 = 0

Via our choice of r(u), this reduces to the

same equation from the previous section:

r(u) = b - Ku

r(u) = - K

r(u) = bu-Kuu

K

Kubu

u

uuu

00

0

00

r

rsol

0u 0

bKK

b0u

1sol

usol

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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 only 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

3) Find the minimum along that vector

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

2) Initially, find the gradient vector

1) Start here

4) Find the conjugate gradient vector*

5) Repeat 3-4 until converged

*) See e.g. Wikipedia, or Scientific Computing by Michael Heath

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Condition number matters for iterative methods

Numerical error becomes significant

for ill-conditioned matrices

A matrix with a condition number of 1

will converge in one iteration, (rare!)

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 course

All these methods make use of PRECONDITIONERS

The system equation, Ku = b, is multiplied by a preconditioner 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 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

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, SLOW)

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 mistake 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 > Chosen study > Direct

Solver selection for 3D problems

Set up a linear problem first

Use the default solver settings for the physics you are solving

If you run out of memory, upgrade to 64-bit and more RAM

Monitor the memory requirements as you grow the problem size

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 only 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

Solutions within

Relative tolerance

Solutions within

Absolute tolerance

t

u

t

u

True solution

t

u At solutions near zero,

the relative tolerance

cannot be used, since it

would result in very

small timesteps

t

u

Tighten both the

absolute and relative

tolerance until you

are satisfied with the

results over time.

Other timestepping options

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

Strict - force the time

stepping method to take

steps that end at the times

specified

Selecting the physics at

startup will usually load

the appropriate settings.

For wave type problems,

Generalized-alpha is

often better.

BDF order, set max and

min to 2 and 1 for wave

problems.

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More timestepping options

Give a condition to stop

at, when this expression

becomes negative

Specify for which steps

the solutions should be

stored

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:

Initial conditions for fluid flow: u = 0

Velocity/Pressure inlet condition

that does NOT ramp up from zero

Solution:

Use flc1hs function to ramp load

and to keep db/dt(t=0)=0

t

u

t

b

u(tinitial) = [Ksteady-state]-1b(tinitial)

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