Introduction to Model Order Reduction

1. Introduction to Model Order Reduction I.2.b Assembling Models from Partial Differential Equation Solvers Luca Daniel Thanks to Jacob White, Deepak Ramaswamy, Michal Rewienski, and Karen Veroy, Jaime Peraire and Tony Patera

2. Partial Differential Equation Solvers • Finite Element Method (FEM) • Heat Conducting Bar example • The “hat” basis functions • The Galerkin scheme • The FEM linear system • An FEM Example • Finite Difference Methods • Heat Conducting Bar • Comparing FEM and F-D in 1-D • 3-D Problems

3. 1-D Example PDE solvers Incoming Heat Far End Temperature Near End Temperature Unit Length Rod Question: What is the temperature distribution along the bar T x

4. Heat flow example PDE solvers Normalized 1-D Equation Normalized Poisson Equation

5. Test Functions The two key steps! Using Basis Functions Partial Differential Equation form Step 1: Choose Basis Functions to represent the solution Step 2: Generate equations for the basis functions weights setting residual orthogonal to some test functions

6. Step1. Basis Functions • Basis for vector space • Basis functions for functional vector space • Examples • exponentials • cos, sin • piecewise constant • piecewise linear

7. Step 1. Basis Functions The basis functions define a space Example “Hat” basis functions Piecewise linear Space

8. Step 2. PDE solvers Galerkin Scheme Force the residual to be “orthogonal” to the basis functions Generates n equations in n unknowns

9. Basis Weights PDE solvers Galerkin with integration by parts Only first derivatives of basis functions

10. Partial Differential Equation Solvers • Finite Element Method (FEM) • Heat Conducting Bar example • The “hat” basis functions • The Galerkin scheme • The FEM linear system • An FEM Example • Finite Difference Methods • Heat Conducting Bar • Comparing FEM and F-D in 1-D • 3-D Problems

11. Basis Weights PDE solvers The FEM linear system

16. Basis Weights FEM The FEM linear system

17. (SPD=Symmetric Positive Definite)

18. Gershgorin Circle Theorem Theorem Statement Picture of Gershgorin Eigenvalues are in the union of all the disks ith circle radius ith circle center

20. Partial Differential Equation Solvers • Finite Element Method (FEM) • Heat Conducting Bar example • The choice of basis functions • The Galerkin scheme • The FEM linear system • An FEM Example • Finite Difference Methods • Heat Conducting Bar • Comparing FEM and F-D in 1-D • 3-D Problems

24. Partial Differential Equation Solvers • Finite Element Method (FEM) • Heat Conducting Bar example • The “hat” basis functions • The Galerkin scheme • The FEM linear system • An FEM Example • Finite Difference Methods • Heat conducting bar: our “informal” Finite Difference • Comparing FEM and F-D in 1-D • 3-D Problems

25. Heat Conducting Bar Demonstration Example Lamp Input of Interest Output of Interest

26. 2) Assign each cut a temperature 1-D Example Heat Flow Discrete Representation 1) Cut the bar into short sections

27. 1-D Example Heat Flow Equation Formulation Incoming heat per unit length Heat in from left Heat out from right Limit as the sections become vanishingly small

28. Finite Difference approximation

29. Heat Conducting Bar Heat In Demonstration Example Basic Equations • Temperature Differential Equation • Spatial Discretization (except at end)

30. Heat Conducting Bar Heat In Demonstration Example Input-Output Discrete Equations

31. Heat Conducting Bar Heat In Demonstration Example State-Space Description Given the right scaling

32. Partial Differential Equation Solvers • Finite Element Method (FEM) • Heat Conducting Bar example • The “hat” basis functions • The Galerkin scheme • The FEM linear system • An FEM Example • Informal Finite Difference Methods • Heat Conducting Bar • Comparing FEM and F-D in 1-D • 3-D Problems

33. FD and FEM (hat basis) Comparing 1D problem FEM FD

34. Partial Differential Equation Solvers • Finite Element Method (FEM) • Heat Conducting Bar example • The “hat” basis functions • The Galerkin scheme • The FEM linear system • An FEM Example • Informal Finite Difference Methods • Heat Conducting Bar • Comparing FEM and F-D in 1-D • 3-D Problems

35. Structural Analysis of Automobiles • Equations • Force-displacement relationships for mechanical elements (plates, beams, shells) and sum of forces = 0. • Partial Differential Equations of Continuum Mechanics

36. Drag Force Analysis of Aircraft • Equations • Navier-Stokes Partial Differential Equations.

37. Engine Thermal Analysis Picture from www.adina.com • Equations • The Poisson Partial Differential Equation.

38. 2-D Discretized Problem FD Matrix properties Discretized Poisson - -

39. 2-D Discretized Problem FD Matrix properties Matrix Nonzeros, 5x5 example

40. 3-D Discretization FD Matrix properties Discretized Poisson - - -

41. 3-D Discretization FD Matrix properties Matrix nonzeros, m = 4 example

42. Summary FD Matrix properties Numerical Properties Matrix is Irreducibly Diagonally Dominant Matrix is symmetric positive definite Assuming uniform discretization, diagonal is

43. : : : Summary FD Matrix properties Structural Properties Matrices in 3-D are LARGE 100x100x100 grid in 3-D = 1 million x 1 million matrix Matrices are very sparse Nonzeros per row Matrices are banded = b = b = b

44. Summary Assembling Models from Partial Differential Equation Solvers • Finite Element Method (FEM) • Heat Conducting Bar example • The “hat” basis functions • The Galerkin scheme • The FEM linear system • An FEM Example • Finite Difference Methods • Heat Conducting Bar • Comparing FEM and F-D in 1-D • 3-D Problems

45. Course Outline Numerical Simulation Quick intro to PDE Solvers Quick intro to ODE Solvers Model Order reduction Linear systems Common engineering practice Optimal techniques in terms of model accuracy Efficient techniques in terms of time and memory Non-Linear Systems Parameterized Model Order Reduction Linear Systems Non-Linear Systems Today Tomorrow Wednesday Thursday Friday