About Weak Form Modeling

Transcription

1 Weak Form Modeling

2 About Weak Form Modeling Do not be misled by the term weak; the weak form is very powerful and flexible. The term weak form is borrowed from mathematics. The distinguishing characteristics of the weak form in COMSOL Multiphysics are that it makes it possible to: Enter certain equations that can be derived from an energy principle in a very compact and convenient form. Such equations, for example, arise in structural mechanics. Add and modify nonstandard constraints, such as various contact and friction models. Build models with extra equations on boundaries, edges, and points. Use the test operator to conveniently work with problems in variational calculus and parametric optimization.

3 About Weak Form Modeling COMSOL Multiphysics provides the possibility to add weak form contributions to any physics interface in the model. In addition, add weak constraints, which, for example, provide accurate fluxes and reaction forces. By default, COMSOL Multiphysics converts all models to the weak form before solving. Although this conversion is a part of the solution technique rather than part of the modeling process, it nonetheless belongs in this discussion because it is based on the weak form implementation. The mathematical weak form gives you direct access to the terms of the weak equation and provides maximum freedom in defining finite element problems.

4 Deriving the Weak Form Consider a stationary PDE problem for a single dependent variable,, in two space dimensions: Γ Ω Now let be an arbitrary function on, and call it the test function ( should of course belong to a suitably chosen well behaved class of functions, ). Multiplying the PDE with this function and integrating leads to Γ where is the area element.

5 Deriving the Weak Form Now use Green s formula (Gauss formula) to integrate by parts: Γ where is the length element. Γ Then use the Neumann boundary condition Γ to arrive at the following equation: 0 Γ Together with the Dirichlet condition, this is a weak reformulation of the original PDE problem.

6 Deriving the Weak Form The requirement is that the above weak equation must hold for all test functions. The names weak and strong stem from this distinction: the weak formulation is a weaker condition on the solution than the strong formulation. A benefit of the weak formulation is that it requires less regularity of Γ. This is important in the finite element method.

7 Entering a PDE in Coefficient Form Using the Weak Form Coefficient Form PDE Ω Ω Ω Derive the weak equation for a coefficient form problem: 0

8 Entering a PDE in Coefficient Form Using the Weak Form Derive the weak equation for a coefficient form problem: 0 Suppose is a scalar. Using the weak form for a 2D problem with a dependent variable, follow these steps: 1. In a field for a weak expression in the Settings windows for a Weak Form PDE node or a Weak Contribution node on a domain, type test(ux)*( c*ux alx*u+gax)+test(uy)*( c*uy aly*u+gay)+ test(u)*(f bex*ux bey*uy a*u)

9 Entering a PDE in Coefficient Form Using the Weak Form 2. In a field for a weak expression in the Settings windows for a Weak Form PDE node or a Weak Contribution node on a boundary, type test(u)*( q*u+g) The variables alx, aly, bex, bey, gax, and gay represent the components of,, and, respectively. You must replace the coefficients in this weak form expression (for example, c, alx, and q) with appropriate expressions. COMSOL Multiphysics automatically adds the Lagrange multiplier.

10 Variables The variables are:, meaning the derivative of with respect to., meaning a second derivative., the tangential derivative variable, meaning the component of the tangential projection of the gradient., meaning component of the tangential projection of the second derivative.