1. NBCR Summer Institute 2006: Multi-Scale Cardiac Modeling with Continuity 6.3 Wednesday: Finite Element Discretization and Anatomic Mesh Fitting Andrew McCulloch and Fred Lionetti

2. Evolved first from the matrix methods of structural analysis in the early 1960’s Uses the algorithms of linear algebra Later found to have a more fundamental foundation The essential features are in the formulation There are two alternative formulations that are broadly equivalent in most circumstances –Variational formulations, e.g. the Rayleigh-Ritz method –Weak or weighted residual formulations, e.g.the Galerkin method Both approaches lead to integral equations instead of differential equations (the strong form) The Finite Element Method

3. Solution is discretized using a finite number of functions –Piecewise polynomials (elements) –Continuity across element boundaries ensured by defining element parameters at nodes with associated basis functions,  1213 14 15 21 22 23 24 FE equations are derived from the weak form of the governing equations R  = 0  Finite differences: Finite elements: R = 0

4. The Finite Element Method Integrate governing equations in each element Assemble global system of equations by adding contributions from each element 1 2 56 7 8 3 4 Element equations 1213 14 15 21 22 23 24 Global equations 

5. Consider the strong form of a linear partial differential equation, e.g. 3-D Poisson’s equation with zero boundary conditions: On region R on boundary S Strong FormLu = f Variational Principle, e.g. minimum potential energy Weighted Residual (weak) form, e.g. virtual work Integral Formulations

6. On region S on boundary C Weak form Integrate by parts Where, u and w vanish at the boundary 00 Weak Form for 2-D Poisson’s Equation

7. Choose a finite set of approximating (trial) functions,  i (x,y), i = 1, 2, …, N Allow approximations to u in the form U(x,y) = U 1  1 + U 2  2 + U 3  3 + … + U N  N (that can also satisfy the essential boundary conditions) Solve N discrete equations for U 1, U 2, U 3, …, U N Galerkin’s Method for 2-D Poisson’s Equation

8. [K]U = F [K] is the stiffness matrix and F is the load (RHS) vector [K] is symmetric and positive definite Galerkin’s Method for 2-D Poisson’s Equation

9. Galerkin is more general than Rayleigh-Ritz. If we add  u/  x, symmetry & the variational principle are lost, but Galerkin still works If w is chosen as Dirac delta functions at N points, weighted residuals reduces to the collocation method If w is chosen as the residual functions Lu-f, weighted residuals reduces to the least squares method By choosing w to be the approximating functions, Galerkin’s method requires the error (residual) in the solution to be orthogonal to the approximating space. The integration by parts (Green-Gauss theorem) automatically introduces the Neumann (natural) boundary conditions The Dirichlet (essential) boundary conditions must be satisifed explicitly when solving [K]U=F Since discretized integrals are sums, contributions from many elements are assembled into the global stiffness matrix by addition. The Ritz-Galerkin FEM finds the approximate solution that minimizes the error in the energy Comments on Galerkin’s Method

10. 1.Formulate the weighted residual (weak form) 2.Integrate by parts (or Green-Gauss Theorem)  reduces derivative order of differential operator  naturally introduces derivative (Neumann) boundary conditions, e.g. flux or traction. Hence called that natural boundary condition 3.Discretize the problem  discretize domain into subdomains (elements)  discretize dependent variables using finite expansions of piecewise polynomial interpolating functions (basis functions) weighted by parameters defined at nodes Steps in the Finite Element Method

11. 4.Derive Galerkin finite element equations  substitute dependent variable approximation in weighted residual integral  Choose weight functions to be interpolating functions — the Galerkin assumption (Galerkin, 1906) 5.Compute element stiffness matrices and RHS  integrate Galerkin equations over each element subdomain  integrate right-hand side to obtain element load vectors which also include any prescribed Neumann boundary conditions Steps in the Finite Element Method (…cont’d)

12. 6.Assemble global stiffness matrix and load vector  Add element matrices and RHS vectors into global system of equations  Structure of global matrix depends on node ordering 7.Apply essential (i.e. Dirichlet) boundary conditions  at least one is required (essential) for a solution  prescribed values of dependent variables at specified boundary nodes, e.g. prescribed displacements  eliminate corresponding rows and columns from global stiffness matrix and transfer column effects of prescribed values to Right Hand Side   the constraint reduced system Steps in the Finite Element Method (…cont’d)

13. 8.Solve global equations  for unknown nodal dependent variables  using algorithms for Ax = b or Ax = x 9.Evaluate element solutions  interpolate dependent variables  evaluate derivatives, e.g. fluxes  derived quantities, e.g. stresses or strain energy  graphical visualization; post-processing 10.Test for convergence  refine finite element mesh and repeat solution Steps in the Finite Element Method (…cont’d)

14. 1234 U 1 =0 2 4 6 8 x u U 4 =9 U 3 =? U 2 =? Galerkin FEM: Simple 1-D Example

15. 2.Integrate by parts (or Green-Gauss Theorem) 1.Formulate the weighted residual (weak) form

16. 4 global nodal parameters U 1, U 2, U 3, U 4 3 linear elements each with 2 element nodal parameters u 1, u 2. Adjacent elements share global nodal parameters, e.g., global parameter U 2 is element parameter u 2 of element 1 and u 1 of element 2. Two (linear) element interpolation functions for each element,  i (x), i = 1, 2 Allow element approximations to u in the form u(x) = u 1  1 + u 2  2 = u i  i i=1,2 3.Discretize the problem

17. 00.51 0 1 x 22 11 element basis functions Element Basis Functions

18. In each element, let u(x)  u 1  1 + u 2  2 = u i  i (x) and w(x)   i (x) 4.Derive Galerkin equations for each element

19. e.g. for Element 1 (no derivative boundary conditions): [k] = [(k ij )] is the element stiffness matrix f = (f i ) is the element load vector 4.Derive Galerkin equations for each element (… cont’d)

20. [k]u = f Element stiffness matrix, [k] and load (RHS) vector, f 5. Compute element stiffness matrices

21. In this problem, each element is the same size and thus: [k] (ele 1) = [k] (ele 2) = [k] (ele 3) and: f (ele 1) = f (ele 2) = f (ele 3) 5. Compute element RHS matrices

22. 6. Assemble global stiffness matrix and load vector

23. That leaves global equations 2 and 3 7. Apply essential (i.e. Dirichlet) boundary conditions

24. Exact! 8. Solve global equations (constraint-reduced)

25. Representing a One-Dimensional Field Polynomials are convenient, differentiated and integrated readily For low degree polynomials this is satisfactory If the polynomial order is increased further to improve the accuracy, it oscillates unacceptably Divide domain into subdomains and use low order piecewise polynomials over each subdomain – called elements

26. Making Piecewise Polynomials Continuous constrain the parameters to ensure continuity of u across the element boundaries or better, replace the parameters a and b in the first element with parameters u 1 and u 2, which are the values of u at the two ends of that element: where is a normalized measure of distance along the curve

27. u = u(x) + + + + + + + + + + + + + x u u = a + bx u = c + dxu = e + fx

28. Linear Lagrange Interpolation

29. Global-Element Mapping Associate the nodal quantity u n with element node n Map the value U  defined at global node  onto local node n of element e by using a connectivity matrix  ( n, e ), Thus, in the first element with u 1 =U 1 and u 2 =U 2.. In the second element u is interpolated by With u 1 =U 2 and u 2 =U 3.

30. We have u (  ) but to define u (x) we need x (  ). Define x as an interpolation of nodal values, e.g. Isoparametric Interpolation   u x u1u1 u2u2 x2x2 x1x1 1 1 u1u1 u2u2 u x2x2 x1x1 x

31. Quadratic Lagrange Basis Functions Use three nodal parameters u 1, u 2 and u 3 are the quadratic Lagrange basis functions.    0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 11 22 33

32. Cubic Hermite Basis Functions 1 1 1 1 1 1 0 0 0 0    

33. Scaling Factors  =0  =1  =0 s1s1 s2s2 s3s3 Global to local mapping: Scaling Factors  arc lengths arc length

34. Two-Dimensional Tensor-Product Elements Bilinear interpolation can be constructed where

35. 1 0 1  1  2 1 1  1  2 u y x  1 x =  n x n u =  n u n y =  n y n 0 Bilinear Tensor-Product Basis Functions

36. A Six-Noded Quadratic-Linear Element 11 22 1.0 0 0 0.5

37. Three-dimensional Linear Basis Functions e.g. trilinear element has eight nodes with basis functions: 1 2 3 4 5 6 7 8 11 22 33

38. 1 2 3 5 6 7 11 22 33 In each node we define: Tri-Cubic Basis Functions

39. Tri-Cubic Basis Functions (Cont’d)

40. Scaling Factors  =0  =1  =0 s1s1 s2s2 s3s3 Global to local mapping: Scaling Factors  arc lengths arc length

41. Coordinate Systems Rectangular Cartesian global reference coordinate system Orthogonal curvilinear coordinate system to describe geometry and deformation Curvilinear local finite element coordinates Locally orthonormal body coordinates define material symmetry and structure, related to the finite element coordinates by a rotation about the -normal axis through the "fiber angle", From Costa et al, J Biomech Eng 1996;118:452-463

42. Curvilinear World Coordinates D) Prolate Spheroidal Coordinates (  )

43. Fiber/Sheet Coordinates

44. Coordinate System Notations

45. Fitting with Linear Lagrange 1-D Elements Two linear Lagrange elements fit the data with a root-mean-squared-error (RMSE) of 0.614892. Result of twice refining the mesh (yielding 8 elements) and re-fitting: RMSE = 0.0930764

46. The least squares fit minimizes the objective function: whereis measured coordinate or field variable; are smoothing weights is the interpolated value at Least Squares Fitting are weights applied to the data points

47. Fitting a Coronary Vascular Tree with Quadratic Lagrange 1-D Elements

48. anesthetized & ventilated New Zealand White rabbit heart arrested in diastole, excised pulmonary vessels removed, aorta cannulated heart suspended in Ringers lactate, perfused in unloaded state with buffered formalin at 80 mm Hg for 4 minutes heart cast in polyvinylsiloxane plunger tube knife heart cast in rubber Rabbit Ventricular Anatomy

49. plunger knife Rabbit Ventricular Anatomy BASE APEX

51.     data point projects onto surface at  d  d  d  Bicubic Hermite isoparametric interpolation (  1,  2 )  { i  i 1 i  4  (  1,  2 )   i  1  i 2 (  1,  2 )   i 2  i 3 (  1,  2 )   2 i  1 2  i 4 (  1,  2 ) } 1 x = d cosh cos  y = d sinh sin  cos  z = d sinh sin  sin 

55.  endo   epi  RIGHT VENTRICLE LEFT VENTRICLE

56. 8,351 geometric points 14,368 fiber angles 36 elements 552 geometric DOF RMSE = ±0.55 mm 184 Fiber angle DOF RMSE = ±19° Anatomic Model Vetter & McCulloch Prog Biophys & Mol Biol 69(2/3):157 (1998)

57. Strain Analysis X c, crossfiber X f, fiber X r, radial

58. A/P View Lateral View Reconstructed 3D Coordinates Transform

59. Baseline2 minutes ischemia End-Systolic Circumferential Strain 0.04 0.00 -0.04 -0.07

60. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 RMS Fitting Error (mm) 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 10 0 Smoothing Weight 

62. Fiber StrainCross-fiber StrainMyocardial Blood Flow Control LAD Occlusion -0.05 0.00 0.05 0.0 1.5 3.0 mL/min/g

64. SEPTAL LATERAL 3months post-surgery Pre-surgery

65. Base Bead Apex Bead 3 Columns of radiopaque beads

66. C L R Undeformed state Deformed state

67. Three-Dimensional Strain Analysis