Model Anything

Quantity Conserved c  advect  diffuse S ConservationConstitutiveGoverning Mass, M  q -- M Momentum fluid, Mv -- F Momentum fluid in pores, Mv -- -- Reactions, M s -- R Dissolved mass in fluid, M s R Heat energy, E S Momentum solid, Mv -- F Viscous fluid Navier Stokes Slow, steady Pore Average Darcy’s Law Mass conservation Mass action Mass action kinetics Advect, dispersion, rxn Convect, conduction Navier Hooke’s Law, elastic solid Dispersion Equilib. sorption Natural convection

Finite Element Method

Approach 1.Mesh Generation: Discretize region into nodes and elements 2.Local Approximation: Assume the result can be approximated locally by interpolating over elements using basis functions. 3.Governing equation Write gov eq. in weak form. Set gov. eq. equal to zero, and sub in the approximation of the result. This gives residual eqn. at each element. 4.Solution Adjust values at nodes to minimize the residual so it is less than a threshold value.

Mesh Generation Element Shape Element Size Mesh Quality Boundary

Basis Functions

ixf(x) Approximating Function Lagrange Polynomial points 3 pts=3 eqs, you can solve this

Interpolation Using Basis Functions Basis Functions

Extend to Multiple Pts 1 value of c at each node 1 Basis function for each node. Basis function only has value from host node to neighbor The type of finite element determines the basis function. The default in many cases is a quadratic element, as above

Governing Eqn

Integration by parts

Weak form in 3D General weak form eqn

v = test function Weak Form Weak Form of Example Governing Eqn Integration by parts Example gov. eqn Equate to u and v Integrate Substitute and use

Use Approximating Functions W is a weighting function. Integrating the weighting function times the residual over the domain equals 0 Galerkin Method W=N

Solving Governing Equation n equations and n unknowns Solve using MUMPS, PARDISO, SPOOLES, other Implemented as Direct Solvers 1.Multiply by Test function and integrate to get weak form. Spatial derivative 1 lower order 2.Approximate functions for h and v 3.Integrate derivative of basis functions locally 4.Get equation for each node 5.Solve simultaneously

Points Galerkin FE with weak form of governing equation Basis functions  element shape functions 2 nd order Lagrange default. 3 nodes along each side. Nodes shared, 2 per element. Degrees of freedom = number of nodes = 2x number of elements for 2 nd order shape function Degrees of freedom sets the size of the problem. Many other types of elements available. First- order=linear basis functions Some problems cannot be solved using one matrix  multiple unknowns, turbulent flow (P and v). Segregate into groups of unknowns, iterate between

Conservation of Mass c =  S e Storage Advective Flux No Diffusive Flux Source Governing  fluid density  porosity S e : degree of saturation