NBCR Summer Institute 2007: Multi-Scale Cardiac Modeling with Continuity 6.3 Wednesday: Finite Element Discretization and Anatomic Mesh Fitting Andrew McCulloch, Stuart Campbell and Fred Lionetti

Wednesday Part I: Steps in the Finite Element Method Part II: Finite Element Interpolation and Basis Functions Part III: Least Squares Geometric and Field Fitting

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 Part I: Steps in the Finite Element Method

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)

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)

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)

1234 U 1 = x u U 4 =9 U 3 =? U 2 =? Galerkin FEM: Simple 1-D Example

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

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

x 22 11 element basis functions Element Basis Functions

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

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)

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

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

6. Assemble global stiffness matrix and load vector

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

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

Part II: Finite Element Interpolation 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 Approximating a 1-D field

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

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

Linear Lagrange Interpolation

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.

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

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

Cubic Hermite Basis Functions    

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

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

1 0 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

A Six-Noded Quadratic-Linear Element 11 2

Three-dimensional Linear Basis Functions e.g. trilinear element has eight nodes with basis functions: 11 22 33

11 22 33 In each node we define: Tri-Cubic Basis Functions

Tri-Cubic Basis Functions (Cont’d)

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

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:

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

Fiber/Sheet Coordinates

Coordinate System Notations

Two linear Lagrange elements fit the data with a root-mean-squared-error (RMSE) of Result of twice refining the mesh (yielding 8 elements) and re-fitting: RMSE = Part III: Least Squares Fitting

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

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

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

plunger knife Rabbit Ventricular Anatomy BASE APEX

    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 

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

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)

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

A/P View Lateral View Reconstructed 3D Coordinates Transform

Baseline2 minutes ischemia End-Systolic Circumferential Strain

RMS Fitting Error (mm) Smoothing Weight 

Fiber StrainCross-fiber StrainMyocardial Blood Flow Control LAD Occlusion mL/min/g

SEPTAL LATERAL 3months post-surgery Pre-surgery