Finite Element Modeling and Analysis with a Biomechanical Application

1. Finite Element Modeling and Analysis with a Biomechanical Application Alexandra Schönning, Ph.D. Mechanical Engineering University of North Florida ASME Southeast Regional XI Jacksonville, FL April 8, 2005

2. Finite Element Modeling The process Elements and meshing Materials Boundary conditions and loads Solution process Analyzing results Biomechanical Application Objective Need for modeling the human femur Data acquisition Development of a 3-Dimensional model Data smoothing NURBS Finite element modeling Initial analysis Discussion and future efforts Presentation overview

3. Finite Element Modeling (FEM) • What is finite element modeling? • It involves taking a continuous structure and “cutting” it into several smaller elements and describing each of these small elements by simple algebraic equations. These equations are then assembled for the structure and the field quantity (displacement) is solved. • In which fields can it be used? • Stresses • Heat transfer • Fluid flow • Electromagnetics

4. Determine the displacement at the material interfaces Simplify by modeling the material as springs. Co F2 = 20kN F3 = 30kN n3 n1 n2 k1 k2 F2 = 20kN F3 = 30kN FEM: The process St

5. F3 Spring force2 = k2(x3-x2) n3 n1 n2 k1 k2 F2 n2 F2 = 20kN F3 = 30kN n3 n1 Spring force2 = k2(x3-x2) Spring force1 = k1(x2-x1) R Spring force1 = k1(x2-x1) FEM: The process • Draw a FBD for each node, sum the forces, and equate to zero ΣF = 0: -k2(x3-x2)+F3 = 0 k2*x2-k2*x3+F3 = 0 -k2*x2+k2*x3 = F3 ΣF = 0: -k1(x2-x1)+k2(x3-x2)+F2 = 0 -k1*x1+(k1+k2)*x2-k2*x3 = F2 ΣF = 0: R+k1(x2-x1)= 0 k1*x1-k1*x2 = R

6. n3 n1 n2 k1 k2 F2 = 20kN F3 = 30kN FEM: The process • Re-write equations in matrix form k1*x1-k1*x2 = R (node 1) -k1*x1+(k1+k2)*x2-k2*x3 = F2 (node 2) -k2*x2+k2*x3 = F3 (node 3) Stiffness matrix [K] Displacement vector {δ} Load vector {F}

7. Apply boundary conditions and solve At left boundary Zero displacement (x1=0) Simplify matrix equation Plug in values and solve n3 n1 n2 k1 k2 F2 = 20kN F3 = 30kN FEM: The process k1=40 MN/m k2 = 60 MN/m x3 x3 x3

8. FEM: The process • The continuous model was cut into 2 smaller elements • An algebraic stiffness equation was developed at each node • The algebraic equations were assembled and solved • This process can be applied for complicated system with the help of a finite element software

9. 1-dimensional Rod elements Beam elements 2-dimensional Shell elements 3-dimensional Tetrahedral elements Hexahedral elements Special Elements Springs Dampers Contact elements Rigid elements Each of the elements have an associated stiffness matrix Different degrees of freedom (DOF) in each of the elements Spring developed has 1 DOF Beam has 6 DOF Linear, quadratic, and cubic approximations for the displacement fields. FEM: Element types

10. Properties Modulus of elasticity (E) Poisson’s ratio () Shear modulus (G) Density Damping Thermal expansion (α) Thermal conductivity Latent heat Specific heat Electrical conductivity Isotropic, orthotropic, anisotropic Homogeneous, composite Elastic, plastic, viscoelastic Strain (%) FEM: Materials

11. n3 n1 n2 k1 k2 F2 = 20kN F3 = 30kN FEM: Boundary Conditions (constraints and loads) • Boundary conditions are used to mimic the surrounding environment (what is not included in your model) • Simple example: Cantilever beam • Beam is bolted to a wall and displacements and rotations are hindered. • More complex example: Tire of a car • Is the bottom of the tire fixed to the ground? • Is there friction involved? • How is the force transferred into the tire? • Are the transfer characteristics of the bearings considered? • Are breaking loads considered? • Interface between components? • Garbage in – garbage out… …but not in FEM • Garbage in – beautiful, colorful, and believable… …garbage out

12. FEM: Solution process • Today’s computer speeds have made FEM computationally affordable. What before may have required a couple of days to solve may now take only an hour. • Inverse of the stiffness matrix • K*δ = F  δ = K-1*F • Displacements  strains  stress

13. FEM: Analyzing results • Interpreting results • Consider the results wrong until you have convinced your self differently. • Sanity checks • Does the shape of the deformation make sense? • Check boundary condition configurations • Are the deformation magnitudes reasonable? • Check load magnitudes and unit consistency • Is the quality of the stress fringes OK? • Smoothness of unaveraged and noncontinuous reslts • Review mesh density and quality of elements • Are the results converging? Is a finer mesh needed? • Verification of results • Local unexpected results may be OK • FBD, simplified analysis, relate to similar studies. • Check reaction forces and moments Pedestal assembly

14. FEM: summary • Use of FEM • Predict failure • Optimize design • The process • Elements and meshing • Materials • Boundary conditions and loads • Solution process • Analyzing results

15. FEM: A biomechanical application • Objective: Develop a high fidelity finite element model of a standard femur • Hexahedral elements • Orthotropic material properties • Macro-inhomogeneous • Realistic loads • Make model available to other researchers

16. Need for Modeling the Human Femur and Adjacent Bones • Improved treatment options for patients with different types of diseases • Legg-Calve-Perthes disease • Osteoporosis • Implant design • Improve implant life and understand failure mechanism • Basic research: Understand the stress distribution in the bone (lower limb) to learn what effect it has on disease and how we can stop or reduce effect of disease or deformities

17. Image Processing • Computed Tomography (CT) data acquisition • Scanning device completes a 360o revolution • Slices are 1 to 5 mm apart (generally). 1mm for male. • Result: Matrix with gray scaled pixels based on tissue density

18. Slice distance Computed Tomography Data Scanning the object Resulting Image Set

19. Select the desired region … and Grow Computed Tomography Data

20. Development of a 3-dimensional model in Mimics Computed tomography data Density threshold in Hounsfield units Cortical : 2000-3200 Cancellous: (1100-2000) Bone Marrow: <1100 Manual editing Region growing

21. Development of a 3-Dimensional model • 3-D models created by interpolation of 2D slices in Mimics • Data smoothing • NURBS (Non-Uniform Rational B-Splines) • *.igs files

22. Data Smoothing • Why smoothen the data? • Data lost through scanning (Interpolation between 1 mm slices) • Estimate the threshold values • Manual editing • Result: Model has rough surfaces • Goal: Want a model that can easily be meshed yet properly represent the object • For meshing to be performed and in order to solve the model it is necessary to remove some detail (partially created from inaccuracies) • Data smoothing through Geomagic by Raindrop Geomagic

23. Problem geometry • Surfaces are not properly closed • Portion of surfaces are inside out

24. NURBS = Non-Uniform Rational B-Splines What is NURBS? Curves that approximate a surface. From a rough surface of “random” points to a surface that can be expressed as polynomials. It creates an analytical surface that the mesher will better understand. Develops a rectangular grid structure in place of the triangular grid structure. The imported geometry consists of triangular surfaces Hexahedral mesh more easily developed with rectangular surfaces Why NURBS? Consecrated method of going from random to analytical surfaces. Initial try vs. final grid structure Need to prepare for the mesh block structure Preparing the geometry for NURBS

25. NURBS surfaces Subdivided Grid Need to plan for the mesh 90O angles is optimal Creation of the NURBS surface

26. Why hexahedral mesh? Simplicity of mesh Regularity Angle distribution Higher control of mesh Why TrueGrid? Specializes in hexahedral meshing of complicated geometry Allows for easy modification of the final mesh Physical window Geometry Elements and nodes Computational window Block structure Command window Environment window Mesh input file is created Development of the mesh: The TrueGrid Interface

27. Development of a hexahedral mesh in TrueGrid 1 Problem elements • Create block structure • Remove blocks not needed (4) • Define block boundary surfaces (5, 6) • Attach the corners of the block to the geometry (7) • Map blocks’ surfaces to the geometry’s “combined” surfaces (7) • Create the mesh by specifying the node increase in each direction. • BB surfaces need to be consistent • Coincident nodes on BB surfaces must be removed Butterfly mesh 3 2 a b 4 5 8 6 7 Surface 1 projected

28. Development of the Femoral Hexahedral Mesh in TrueGrid • More complicated block structure • Three separate volumes: cortical, cancellous, marrow • Long section – easy • Condyles, trochanter and femoral head causes problems • Visualization difficulty: how to create the block boundaries in 3 dimensions • In some cases it is impossible to ensure the 3D connectivity • Ensuring that blocks will allow for different material properties. • Geometry is free-formed • 90 degree corners don’t exist • Try to map the blocks’ vertices to convex geometry • Must carefully plan (from the beginning) how to map the blocks.

29. *.iges file contains more surfaces than block surfaces needed Surfaces are combined Vertices are attached Edges are attached Curves are generated to steer the mesh Faces are projected Preparing the geometry in TG

30. Blocks missing Blocks missing Developing the block structure

31. Need to ensure that separate blocks are created for cortical, cancelleous, and marrow materials Connectivity between blocks may not be possible in some regions. Create separate files and then merge the files 6 edges Not possible Creating the block structure Solution

32. Lowest level of blocks Excluding cortical shell Block face follows geometry that is partially concave Elements intersect themselves Zero and negative volume Meshing difficulty due to geometry

33. 1 2 Meshing difficulty due to geometry Non controlled edge • Angle 1 is acceptable • Angle 2 is negative • Only edges attached to geometry are initially controlled • Curves and internal surfaces are created to control the mesh

34. Meshing difficulty due to geometry • Resolving geometry difficulties • Go back to Geomagics and recreate the *iges files. Move the edges of the surfaces to a location less likely to create problems (less concave) • Create internal edges, surfaces and points to steer the internal mesh • Use bias commands to increase/decrease the mesh density in a local region • Resolving block structure difficulties • May need to build separate

35. Resulting mesh 1 2 3 4 5 6

36. How can mesh be modified? • Element size • Element material • Insert partitions in the mesh • Insert geometry of an implant • Attach the internal block surfaces/edges/vertices to the implant • Change the material properties of the implant • Merge different mesh files

37. Initial analysis performed to verify mesh solvability Material Linearly elastic Isotropic macroinhomogeneous Boundary conditions Removed all DOF from the distal end of the condyles Load One legged stance Distributed load on femoral head Distributed load on greater trochanter Results Max deflections: 3 mm Peak von Mises stress: 37 MPa F1 F2 z y x Analysis Marco Viceconti, Mario Davinelli, Fulvia Taddei, Angelo Cappello, “Automatic generation of accurate subject-specific bone finite element models to be used in clinical studies”, Journal of Biomechanics 37 (2004) 1597 - 1605

38. Future Work and Discussion • Model improvements • Femur • Orthotropic material properties • Improved loading conditions • Compare to tetrahedral mesh • Analyze different stances • Complete a model of the lower limb • Model use • Optimize implant designs • Improved treatment options for patients with different types of diseases • Make available to the public so research can more easily be advanced

39. Summary • Garbage in – garbage out! • Even though you obtain pretty pictures. • Anyone can run a FE analysis… • Pay close attention to boundary conditions, degrees of freedom, mesh quality and validity of results • Applications • Failure analysis, optimization, heat transfer, fluid flow, electromagnetic analysis • Biomechanical application