1. BENG 276 HHMI Interfaces Lab 2: Numerical Analysis for Multi-Scale Biology Introduction to Finite Element Modeling Andrew McCulloch Department of Bioengineering amcculloch@ucsd.edu

2. Classes of Model Class of ModelBiomechanicsElectrophysiologyBiotransport Boundary conditions Tissue boundary loads Passive tissue current flows Membrane sources and sinks Continuum PDEsEquations of motion Monodomain equation Reaction- diffusion equation Constitutive Model Strain energy formulation Anisotropic conductivities Diffusion model Systems ModelMyofilament activation and interactions Ionic currents and action potential Reaction network model

3. 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

4. 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, Ψ 1213 14 15 21 22 23 24 FE equations are derived from the weak form of the governing equations Finite differences: R = 0 Finite elements: ∫RΨ = 0

5. 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 

6. 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

7. 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

8. Choose a finite set of approximating (trial) functions, Ψ i (x,y), i = 1, 2, …, N Allow approximations to u in the form U(x,y) = U 1 Ψ 1 + U 2 Ψ 2 + U 3 Ψ 3 + … + U N Ψ 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

9. [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

10. 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

11. 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

12. 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)

13. 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)

14. 8.Solve global equations  for unknown nodal dependent variables  using algorithms for Ax = b or Ax = λx 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)

15. 1234 U 1 =0 2 4 6 8 x u U 4 =9 U 3 =? U 2 =? Galerkin FEM: Simple 1-D Example

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

17. 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, Ψ i (x), i = 1, 2 Allow element approximations to u in the form u(x) = u 1 Ψ 1 + u 2 Ψ 2 = u i Ψ i i=1,2 3.Discretize the problem

18. 00.51 0 1 x 22 11 element basis functions Element Basis Functions

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

20. 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)

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

22. 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

23. 6. Assemble global stiffness matrix and load vector

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

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

26. 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

27. 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

28. Linear Lagrange Isoparametric Interpolation u = (1-  )u 1 +  u 2 =  1 u 1 +  2 u 2 x u 0 1  element 1 element 2element 3 nodes u1u1 u2u2 u3u3 u4u4 + + + + + + + + + + + + + 0 1  1     0 1 1    x = (1-  )x 1 +  x 2 =  1 x 1 +  2 x 2   u x u1u1 u2u2 x2x2 x1x1 1 1 u1u1 u2u2 u x2x2 x1x1 x  1 and  2 are the1-D linear Lagrange basis functions

29. 1 0 1  1  2 u y x x =  n x n u =  n u n y =  n y n 1 1  1  2  0 Bilinear Lagrange Tensor-Product Interpolation Bilinear interpolation can be constructed: where  1 –  4 are the 2-D bilinear Lagrange basis functions

30. 3-D Trilinear Lagrange Basis Functions 1 2 3 4 5 6 7 8 11 22 33 The trilinear element has 8 nodes with basis functions formed from the 1-D linear Lagrange functions  1 =1- , and  2 =  :

31. 1-D Quadratic Lagrange Basis Functions Use three nodal parameters u 1, u 2 and u 3    0 0.5 1.0 0 0.5 1.0 0 0.5 1.0 φ1φ1 φ2φ2 φ3φ3  1,  2,  3 are the 1-D quadratic Lagrange basis functions

32. 1-D Cubic Hermite Basis Functions 1 1 1 1 1 1 0 0 0 0    

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

34. A Six-Node Quadratic-Linear Element 11 22 1.0 0 0 0.5

35. 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

36. At each node we need 8 parameters: Tri-Cubic Hermite Basis Functions 1 2 3 4 5 6 7 8 11 22 33

37. Scaling Factors  =0  =1  =0  =1 s1s1 s2s2 s3s3 Global to local mapping: Scaling Factors = arc lengths arc length

38. Numerical Integration

39. Gauss-Legendre Quadrature 2integration points and 2 weights can exactly integrate a polynomial with 4 coefficients (i.e. a cubic) exactly:

40. Three-Point Gaussian Quadrature

41. Three-Dimensional Quadrature Integration

42. External and Material Coordinate Systems Rectangular Cartesian global reference coordinate system (Y 1,Y 2,Y 3 ) Orthogonal curvilinear world coordinate system to describe geometry and deformation (  1,  2,  3 ) Curvilinear local finite element coordinates (  1,  2,  3 ) Locally orthonormal material coordinates define material symmetry and structure (X 1,  X 2,  X 3 ) From Costa et al, J Biomech Eng 1996;118:452-463

45. Ventricular Geometry Base Apex RV LV Epicardium Endocardium Septum RA LA Aorta Pulmonary artery SVC Pulmonic Tricuspid valves Mitral Truncated ellipses of revolution b a h/ah/a Canine left ventricle, from: Streeter DD Jr, Hanna WT (1973) Circ Res 33(6):639-55

46. Prolate Spheroidal Coordinates (  ) x 1 = d cosh  cos  x 2 = d sinh  sin  cos  x 3 = d sinh  sin  sin  a d b    x x 1 2

47. Trilinear Lagrange Interpolation of Prolate Spheroidal Coordinates 0 Y1Y1 Y2Y2 * Unloaded (zero pressure) end-diastolic state Node  10.4*00 20.41200 30.700 4 1200       node 1 node 2 node 3 node 4 d=3.7 cm

48. Myofiber Architecture endocardium midwall epicardium

49. Laminar Sheet Structure x510

50. An example of an orthonormal local material coordinate system (x f, x n, x s ) defined here by two angles, a and b, measured with respect to local finite element coordinates Fiber/Sheet Material Coordinates

51. Costa KD, Takayama Y, McCulloch AD, Covell JW (1999) Am J Physiol 276(2 Pt 2):H595-607 Fiber/Sheet Orientations Fiber Angle (deg) Arts T, Costa KD, Covell JW, McCulloch AD (2001) Am J Physiol Heart Circ Physiol 280(5):H2222-9 Geerts L, Bovendeerd P, Nicolay K, Arts T (2002) Am J Physiol Heart Circ Physiol 283(1):H139-45

52. Linear Fiber/Sheet Angle Parameters NodeFiber angle, ° Transverse angle, ° Sheet angle, ° 1800-30 2,80015 3,-7000 4,-700-45       node 1 node 2 node 3 node 4 d=3.7 cm 0 Y1Y1 Y2Y2