Intro to Finite Element Modeling

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Intro to Finite Element Modeling COMSOL

• (a mini-course)
• CEE-268, Winter 2006
• By Michael Cardiff

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What is a model?

• Straight from the OED
• a system or thing used as an example to follow
  or imitate - a simplified description, especially
  a mathematical one, of a system or process, to
  assist calculations and predictions.

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What is a numerical model?

• A model which estimates the solution to a hard
  problem (usually a set of PDEs) through numerical
  approximations.

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A finite-difference example
?

• Analytical Solution
• Integrate the expression using calculus tricks
• Plug in BCs to get values for integration
  constants

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A finite-difference example

• Numerical Solution
• Approximate derivatives using numerical tricks
• Write all equations into a single matrix and
  solve.

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A finite-difference example
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When should we use numerical methods?

• Necessary Assumptions for the methods we have
  studied so far
•  
• 2-D problems, in plan view generally
• Horizontal-only flow (Dupuit-Forchheimer
  assumption)
• Isotropic, homogeneous domain
• Steady-state conditions (no change in time)
• Darcys law applies in the full domain

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When should we use numerical methods?

• To get approximate solutions to problems that
  cannot be solved analytically, for example
• Problems with complicated geometries and/or
  boundaries
• Problems with difficult to solve PDEs
• To test the applicability of a simple rule under
  a variety of conditions (for example, the
  Ghyben-Herzberg relation)
• To verify the correctness of an analytical
  solution

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Issues to Remember

• It may take a long time to get your solution (or
  you may get a lack of convergence!)
• When/if you get a solution, it is only an
  approximate solution.
• The solutions may be highly dependent on the data
  you give to COMSOL (anisotropy, heterogeneity,
  etc)
• Your data has errors.

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COMSOL

• Based on the Finite Element (FE) Method
• () FE can handle complex geometries and boundary
  conditions with ease. FD is basically restricted
  to rectangular shapes.
• () FD only tries to minimize error at discrete
  points, whereas FE tries to minimize the error
  over the entire element (line segments in 1-D,
  triangles in 2-D)
• () The mathematics behind FE are more involved
  than FD.

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COMSOL
Geometry (CAD)
PDE Definition
COMSOL
Discretization
PDE Solution
Post-Processing
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COMSOL ?Geometry

• Two options
• Simple UI for 1-D, 2-D, and 3-D
• Points, lines, planes, circles, etc.
• Object mathematics, scaling, rotation, reflection
• Extrusion / Revolving in 3-D
• Import from standard CAD software
• VRML, dxf

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COMSOL ?PDE Definition

• Pre-defined physical PDEs (coupled through
  multi-physics)
• OR
• General PDE solution mode

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COMSOL ?PDE Definition

• Define parameter values within each shape
  (subdomain)
• Hydraulic conductivity in aquifer
• Define boundary conditions
• Head at specified boundary
• Define known values at points
• Known extraction rate at point-source well
• Couple physics as necessary
• Solute transport (advection/diffusion equation)
  coupled to fluid velocity (Darcys Law or N-S
  equations)

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COMSOL ?Discretization

• Automatic Meshing Options
• Element shape (triangular, square)
• Element type (Linear, Quadratic, Cubic)
• Mesh parameters (rate of element growth, mesh
  size sensitivity)
• Advanced moving meshes / automatic refinement

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COMSOL ? PDE Solving

• Options
• Transient (time dependent) or Stationary (steady
  state) solving
• Great assortment of FE algorithms
• UMFPACK, GMRES, Multi-grid, Conjugate Gradient
• Can solve individual parts of problems, store
  solutions, or iterate

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COMSOL ? Post-Processing

• For any given physics, many variables will be
  output
• Example Darcys law will have head, velocity,
  pressure as output
• Variety of standard visualizations
• Surface, contour, streamline, arrow, and animated
  plots
• Further analysis
• Subdomain / boundary integration
• Cross-sectional plots

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COMSOL

• Compared with other methods of modeling
• () Relatively easy to use graphical interface
• () Uses state-of-the-art solvers and optimizers.
  Runs well on a suitably-equipped (lots of RAM)
  desktop PC.
• () Lots of default options / hidden parameters
• () Interface changes based on what type of
  physics you are solving for. Multiphysics gets
  even more cumbersome.