1. Equation-Based Modeling
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2. Multiphysics and Single-Physics Simulation Platform
Mechanical, Fluid, Electrical, and Chemical Simulations
Multiphysics - Coupled Phenomena
Two or more physics phenomena that affect each other with no limitation on
which combinations
how many combinations
Single Physics
One integrated environment – different physics and applications
One day you work on Heat Transfer, next day Structural Analysis, then Fluid Flow, and so on
Same workflow for any type of modeling
Enables cross-disciplinary product development and a unified simulation platform

3. Highly Customizable and Adaptable
Create your own multiphysics couplings
Customize material properties and boundary conditions
Type in mathematical expressions, combine with look-up tables and function calls
User-interfaces for differential and algebraic equations
Parameterize on material properties, boundary conditions, geometric dimensions, and more
High-Performance Computing (HPC)
Multicore & Multiprocessor: Included with any license type
Clusters & Cloud: With floating network licenses

4. Product Suite – COMSOL 4.4

5. When is Equation-Based Modeling Needed?
Try to avoid equation-based modeling if possible!
Using built-in physics interfaces enables ready-made postprocessing variables and other tools for faster model setup with much lower risk of human error
Applications that previously required equation-based modeling but now has a dedicated physics interface:
Fluid-Structure Interaction (Structural Mechanics Module, MEMS Module)
Surface adsorption and reactions (Chemical Reaction Engineering Module, Plasma Module)
Shell-Acoustics and Piezo-Acoustics (Acoustics Module)
Thermoacoustics (Acoustics Module)
and many more…

6. When is Equation-Based Modeling Needed?
Try to avoid equation-based modeling if possible!
But: we don’t have every imaginable physics equation built-into COMSOL (yet!). So there is sometimes a need for custom modeling.

7. Custom-Modeling in COMSOL
COMSOL Multiphysics® allows you to model with PDEs or ODEs directly:
Use one of the equation-template user interfaces
You do *not* need to write “user-subroutines” in COMSOL to implement your own equation!
Benefit: COMSOL’s nonlinear solver gets all the nonlinear info with gradients and all. Faster and more robust convergence.

8. Customization Approaches
Four modeling approaches:
Ready-made physics interfaces
First principles with the equation templates
Start with ready-made physics interface and add additional terms.
Start with a ready-made physics interface and add your own separate equation (PDE,ODE) to represent physics that is not already available as a ready-made application mode
Also:
The Physics Builder lets you create your own user interfaces that hides the mathematics for your colleagues and customers

9. Linear Model Problems: Fundamental Phenomena
Laplace’s equation
Heat equation
Wave equation
Helmholtz equation
Convective Transport equation

10. COMSOL PDE Modes: Graphical User Interfaces
Coefficient form
General form
Weak form
All these can be used for scalar equations or systems
Which to use?
Whichever is more convenient for you and your simulation needs

11. Coefficient Form Coefficient Matching Example: Poisson’s equation
inside domain
on boundary
inside subdomain
on subdomain boundary
Implies c=f=h=1 and all other coefficients are 0.

12. Example:
Block: 10x1x1
PDE: default Poisson’s equation with unknown u.
Dirichlet boundary condition everywhere: u=0

13. Model Wizard: Coefficient Form PDE with one dependent field variable u

14. Stationary study

15. Geometry: block 10-by-1-by-1. Units in meters (SI).

16. Coefficient Form PDE with c=1, a=0, f=1

17. Mass Coefficients are inactive due to Stationary Study

18. Dirichlet Boundary Condition

19. All boundaries: u=0

20. Automatic tet mesh
….or swept hex mesh

21. Control over shape function and element order

22. Stationary solution
Plot of dependent field variable u on slices

23. Differentiate u with respect to x: d(u,x)

24. Recover option for derivatives switched on
Recover option for derivatives switched on. Gives smoother derivative field.

25. Also available as ppr operator.
The Recover feature applies “polynomial-preserving recovery” on the partial derivatives (gradients).
Higher-order approximation of the solution on a patch of mesh elements around each mesh vertex.
Also available as ppr operator.
d(u,x) with no Recover smoothing
d(u,x) with Recover smoothing

26. Second derivative: d(d(u,x),z)

27. Coefficient Form, Interpretations
source
diffusion
absorption
convection
source
convection
damping/mass
mass

28. Coefficient Form, Structural Analysis Wave Equation
initial/thermal stress
elastic stress
body force (gravitation)
damped mass
mass
density
damping coefficient
stress, u= displacement vector
stiffness, “spring constant”

29. Coefficient Form, Transport Diffusion Equation
source
diffusion
absorption
convection
source
convection
accumulation/storage

30. Coefficient Form, Steady-State Equation

31. Coefficient Form, Frequency-Response Wave Equation
diffusion
Helmholtz term
source
Helmholtz equation:
Wave number
Wave length

32. Example:
lambda=2.5
k=2*pi/lambda
a= - k^2
f=0
u=1 one end
u=0 other end

33. User-defined Parameters:
Wavelength: lambda=2.5 m
Wavenumber: k=2p/lambda~2.5 m-1

34. Coefficient Form PDE
c=1
a=-k^2
f=0

35. Dirichlet Boundary Condition u=1
Oscillating wave with peak value 1
Spatial frequency is given by wavenumber k

36. Dirichlet Boundary Condition u=0
Mirror reflection

37. Solution u: wave with 4 wavelengths over 10 m block length

38. Text input field allows typing complex valued expressions.
Here: u + superimposed higher frequency wave with wavenumber 5*k

39. abs() for absolute value (complex modulus)

40. Complex Arithmetics
Can compute:
real(w)
imag(w)
abs(w)
arg(w)
conj(w)

41. General Form – A more compact formulation
inside domain
on domain boundary
For Poisson’s equation, the corresponding general form implies
All other coefficients are 0

42. Weak Form
Think of the weak form as a generalization of the principal of virtual work (for those familiar with that) with virtual displacement du
The test function n ~ du
Convection-diffusion equation:
Multiply by test function n and integrate:
Integrate by parts and use boundary conditions:
In COMSOL you can type the integrands of this integral expression: Weak Form PDE

43. Typing the Weak Form c*grad(u) ·grad(test(u))=
c*grad(u) ·test(grad(u))=
c*(ux*test(ux)+uy*test(uy)+uz*test(uz))
Note: COMSOL convention has the integral in the right-hand side so additional negative sign needed in the GUI

44. Modifying Variables and Equations
Enable Equation View on the Model Builder Show menu
Once enabled, Equation View stays enabled for new models
Variables, weak expressions and constraints can be modified
Modified rows are marked with warning signs
Use reset buttons to cancel modifications

45. Transient Diffusion Equation ~ Heat Equation
source
accumulation/storage

46. Example:
c=1
da=1
f=0
Transient 0->100 s
“Cooling” u=0 at ends
“Heat Source” f=5+3*sin(2*pi*0.1[Hz]*t)

47. Time Dependent study

48. 3 overlapping blocks of length 4,6, and 10 m
COMSOL partitions these into 3 non-overlapping domains.
Field and flux automatically continuous across interior boundaries

49. Cofficient Form PDE with no volume source: f=0

50. Cofficient Form PDE with no volume source: f=0

51. Mass Coefficients are here active due to Time Dependent Study

52. Superimposed source term: f=5+3*sin(2*pi*0.1[Hz]*t)

53. u=0 at the ends

54. Time Dependent study settings: solve between 0s and 100s, output solution at every 0.1s.
Underlying time stepping algorithm is automatic and controlled by user-defined tolerances.

55. Solution u at 38.2s

56. Sample solution inside domain using Domain Point Probe

57. Probe position controlled by slider control

58. Value of u vs. time at probe location

59. Equation Systems COMSOL can handle systems of equations in all of
Coefficient Form
General Form
Weak Form
or combinations of the above
Easy setup from Model Wizard

60. Model Wizard: Coefficient Form PDE with two dependent field variables u1 & u2

61. The Coefficient Form PDE for two dependent field variables.
The PDE coefficients and sources are now matrices (or high-order tensor-like entities) and vectors.

62. Coefficient Form PDE for 2 variables in 2D Spaceda c b a f
The default coefficients corresponds to two decoupled Poisson’s equations. Fill out with nonlinear or off-diagonal coefficients, as well as nonlinear source terms for couplings.

63. Examples: Electrical Signals in a Heart, General Form PDE
Fitzhugh-Nagumo Equations
𝜕 𝑢 1 𝜕𝑡 = 𝛻 2 𝑢 1 + α− 𝑢 1 𝑢 1 −1 𝑢 1 − 𝑢 2
𝜕 𝑢 2 𝜕𝑡 =ε 𝛽 𝑢 1 − 𝛾𝑢 1 −𝛿
Landau-Ginzburg Equations
𝜕 𝑣 1 𝜕𝑡 − 𝛻 2 (𝑣 1 − 𝑐 1 𝑣 2 )= 𝑣 1 − 𝑣 1 − 𝑐 3 𝑣 2 𝑣 𝑣 1 2
𝜕 𝑣 2 𝜕𝑡 − 𝛻 2 ( 𝑐 1 𝑣 1 + 𝑣 2 )= 𝑣 2 − 𝑐 3 𝑣 1 − 𝑣 2 𝑣 𝑣 1 2

64. Simplified representation of a heart as ½ sphere + ½ ellipsoid

65. Fitzhugh-Nagumo Equations

66. Fitzhugh-Nagumo Equations
Solution: u1

67. Landau-Ginzburg Equations

68. Landau-Ginzburg Equations
Solution: v1

69. End of Presentation