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

Presentation overview 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

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

FEM: The process Determine the displacement at the material interfaces Simplify by modeling the material as springs. Co F3 = 30kNF2 = 20kN St k1 k2 F3 = 30kNF2 = 20kN n1n2 n3

FEM: The process Draw a FBD for each node, sum the forces, and equate to zero k1 k2 F3 = 30kNF2 = 20kN n1n2 n3 F3 Spring force2 = k2(x3-x2) ΣF = 0: -k2(x3-x2)+F3 = 0 k2*x2-k2*x3+F3 = 0 -k2*x2+k2*x3 = F3 Spring force1 = k1(x2-x1) n2 F2 Spring force2 = k2(x3-x2) ΣF = 0: -k1(x2-x1)+k2(x3-x2)+F2 = 0 -k1*x1+(k1+k2)*x2-k2*x3 = F2 n1 Spring force1 = k1(x2-x1) R ΣF = 0: R+k1(x2-x1)= 0 k1*x1-k1*x2 = R

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} k1 k2 F3 = 30kNF2 = 20kN n1n2 n3

FEM: The process Apply boundary conditions and solve At left boundary  Zero displacement (x1=0) Simplify matrix equation Plug in values and solve k1 k2 F3 = 30kNF2 = 20kN n1n2 n3 k1=40 MN/m k2 = 60 MN/m x3

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

FEM: Element types 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: Materials 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: 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 k1 k2 F3 = 30kNF2 = 20kN n1n2 n3

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

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

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