Introduction to COMSOL Multiphysics

July 9, 2015Yosuke Mizuyama, Ph.D.

COMSOL, Inc.

Introductory tutorial in Kanda

KEYWORDS

Partial differential equations (PDEs)

Multiphysics

Equation-based interface

Application builder

Finite element method (FEM)

Partial differential equations

Almost all physical phenomena can be described by partial differential equations (PEDs).

,

For given find such that Γ

Γ Ω

It is VERY easy to solve this PDE for simple geometries and IC, BCs

Solution

= 0 = 1

= 0

= 0

Fourier’s law

Solution

= 0= 0

= 0

= 0

It is VERY easy to solve this PDE for simple geometries and IC, BCs

Solution

= 0

= 0

It is VERY easy to solve this PDE for simple geometries and IC, BCs

= 1

Solution

= 0

= 0= 0

This geometry is still simple but it is EXTREMELY DIFFICULT to solve this PDE by hand or in your head.

= 0= 1

= 0

This is way beyond a hand calculation! – Non-linear problem.

Solution

= 0 = 1= 0

= 0

In general, almost all cases fall into the class that has “non-trivial” or “non-analytical” solutions.

Then we need numerical methods to solve the problem, which includes FEM, FDM, FVM, BEM, etc.

The finite element method

The domain is meshed into a collection of finite elements . The governing PDEs are satisfied in each local element. The original PDEs (called strong form) are formulated to a weak

form, which reduces the spatial derivative order by 1.

Ω Ω discretization

Weak formulation

Compared to the finite difference method

The domain is tessellated into a collection of finite grid . The spatial derivatives in a strong form is explicitly discretized.

Ω Ωdiscretization

The FEM tries to find solutions in a larger function space for a weak form rather than a narrow one for a strong form on an element.

Function space for

strong form

Function space for weak form

The finite element method

,,

The finite element method

Solutions to the strong form

,,

The finite element methodSolutions to the weak form

The finite element method

The error estimate theory predicts the mesh dependence for the FEM accuracy.

The finite element method

Mesh size

L2 e

rror

The error estimate theory predicts the mesh dependence for the

FEM accuracy.

The finite element method

Pros

Solid mathematical background for the error estimate, convergence and stability established by the functional analysis.

High adaptability for complex geometries.

Natural boundary condition is embedded.

Cons

Needs more memoryMore computation time Needs mathematical

knowledge

Multiphysics

Most physics are multiphysics.

Fluid / structure Electric / heat / structure Ray / heat / structure

Multiphysics

COMSOL provides a general coefficient form

0====== faea βγα

fauuuuctud

tue aa =+∇⋅+−+∇⋅∇−

∂∂+

∂∂ βγα )(2

2

fauuuuctud

tue aa =+∇⋅+−+∇⋅∇−

∂∂+

∂∂ βγα )(2

2

0)( =∇⋅∇− ucud ta

Equation-based interface

Equation-based interface

Type in any expression in math and logical operator

Worldwide Sales Offices