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Finite Element Modeling and Analysis with a Biomechanical Application

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. Finite Element Modeling The process Elements and meshing Materials

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Finite Element Modeling and Analysis with a Biomechanical Application

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  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

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