Parallel Genetic Algorithm

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Simulation and Optimization of Electromagnetohydrodynamic FlowsBrian Dennis Ph.D. Candidate Aerospace Engineering Dept. Penn State University George Dulikravich Professor Mechanical and Aerospace Dept. University of Texas at Arlington


OverviewIntroduction Governing equations for EMHD General LSFEM formulation h and p-type finite element solver LSFEM for EMHD Code validation cases Applications Conclusion and recommendations


IntroductionEMHD is study of incompressible flow under the influence of electric and magnetic fields Various simplified analytical models exist and have been used for numerical simulation(EHD,MHD)


Introduction cont.A fully consistent non-linear model for general EMHD has been recently developed No numerical simulations of full EMHD has been report in open literature to date A few numerical simulations of simplified or inconsistent EMHD models have been report in open literature A computer code for numerical simulation for 2-D planar MHD/EMHD flows has been developed using LSFEM


Introduction cont.Numerical simulation is necessary for performing optimization involving EMHD flows


Applications of EMHDManufacturing(solidification,crystal growth) Flow control Drag reduction/propulsion Pumps with no moving parts(artificial heart, liquid metal pumps) Compact heat exchangers Shock absorbers, active damping


Fully Consistent Model of EMHD


Conservation of Linear Momentum


Conservation of EnergyConservation of Mass


Maxwell’s Equations


Sub-modelsThe LSFEM was applied to two sub-models of the fully nonlinear Electro-magneto-hydrodynamic system Magneto-hydrodynamics Electro-magneto-hydrodynamics with reduced number of source terms


Least-Squares Finite Element Method (LSFEM)


Advantages of the LSFEMCan use equal order basis functions for pressure and velocity can use the first order form of PDE’s can handle any type of equation and mixed types of equations Can discretize convection terms without upwinding or explicit artificial dissipation Stable and robust method Resulting system of equations is symmetric and positive definite


Simple iterative techniques such as PCG and multigrid can be used to solve the system of equations Inclusion of divergence constraint for magnetic flux is straight forward


Least-squares finite element method (LSFEM)


h and p-Type Finite Element Solver


h and p-Type Methods h-type finite element methods use large numbers of low order accurate elements p-type finite element methods use small numbers of high order accurate elements h/p-type finite element methods use a combination of both types together with an adaptive strategy p-type methods convergence to the exact solution is more rapid than in with h-type methods, if the underlying solution is smooth


p-Type Method element approximation function are compose of p-type expansions. These expansions are composed of a summation of P polynomials p-type expansions can be categorized as nodal expansions or modal expansions the unknown coefficients in nodal expansions are the values of the function at the nodes of the element and therefor have some physical meaning. Nodal expansions based on Lagrange polynomials are typically used for low order finite elements the unknown coefficients in modal expansions are not associated with any nodes


p-Type MethodModal expansions are hierarchical, that is, that all expansion sets less than P+1 are contained within the expansion set P+1 Nodal expansions are not hierarchical Modal expansion typically produce finite element matrices that have a lower condition number than those produced with nodal expansions Expansions are usually developed in one dimension. Approximation functions for multidimensional elements are constructed through tensor products of the one dimensional expansions. Typically, the modal expansions are combined with the first order Lagrange polynomials so that continuity of the solution is satisfied between neighboring elements in a mesh


LSFEM CodeA serial code was developed in C/C++ to solve general systems with LSFEM Steady state problems only Mixed triangular and quadrilateral meshes for h-type elements. Quadrilateral elements for p-type elements. Element approximation functions are constructed from a modal basis derived from Jacobi polynomials Support for multiple material domains such as in conjugate heat transfer problems for h-version Nonlinear equations are linearized with Newton or Picard method


LSFEM CodeIntegrals are evaluated numerically using Legendre-Gauss quadrature. A P+2 rule was used for p-type elements. Static condensation is used to remove the interior degrees of freedom from p-type elements.


Sparse Linear SolversDirect Solver: Sparse LU factorization from PETSc library Iterative Solvers: Jacobi preconditioned conjugate gradient method Multi-p multilevel method




Nondimensional Form for EMHD Equations


Nondimensional Form for EMHD Equations

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Last Updated: 8th March 2018