introduction to comsol multiphysics - kesco · introduction to comsol multiphysics july 9, 2015...
TRANSCRIPT
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
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