The University of SydneySlide 1 Simulation Driven Biomedical Optimisation Andrian Sue AMME4981/9981 Week 5 Semester 1, 2016 Lecture 5

The University of SydneySlide 2 Modelling Workflow for Biomedical Implant Verification Solid modelling Finite element analysis Optimisation Data acquisition

The University of SydneySlide 3 Optimisation Framework and Methodology

The University of SydneySlide 4 Overview –How to approach an optimisation problem in implant design? –Framework for optimisation –Response Surface Method –Example: Functionally Graded Dental Implant

The University of SydneySlide 5 Direct Analysis problem Try many different designs Make final decision based on chosen designs Inverse Design problem Find relationship between design and result Determine design from an optimal result Direct and Inverse Approaches

The University of SydneySlide 6 Defining geometry and/or material Discrete design variables Locus of solutions dependent on variable Parametric Material redistribution Discretisation of structure (homogenisation) Solutions dependent on constraints only Topology Types of Structural Optimisation

The University of SydneySlide 7 A Parametric Approach to Implant Optimisation Min stress Max durability Min cost Min stress shielding Geometry Material Properties Stress Strain Displacement Stiffness Dynamic response Geometry Material properties Boundary conditions Loads What does FEM need? What does FEM provide? What is expected from the design? How can we improve the design? “trial and error” validation

The University of SydneySlide 8 Direct Parametric Optimisation of a Total Hip Replacement –Change the design variables – Geometry (Length) – Material (Young’s Modulus) –By FEA, determine biomechanical responses – Peak von Mises stress in implant – Displacement – Interfacial stress (at bone-implant interface) 50 L 20 60º 50 Ball R=15 Rod r=7.5 x y 40 Taper 1:25 F E

The University of SydneySlide 9 A Framework for Optimisation –Optimal objective function –The objective function drives the optimisation Min y 0 (x), where y 0 (x) is the objective function –Design constraints –Limiting conditions y j (x) ≤ 0, where j = 1,2,…,n c –Design variable vector –Selected design variables x = [ x 1, x 2, …, x N ] T –Design space –Available values of the design variable x l ≤ x ≤ x w where x is a design variable

The University of SydneySlide 10 Design Variable Selection –There can be more than one –E.g. scaffold design –Periodic structure –Consider one unit cell –Four design variables –Diameter of fibres –Spacing in x direction –Spacing in y direction –Young’s Modulus of fibres x 1 =d x2=gxx2=gx x3=gyx3=gy x 4 =E

The University of SydneySlide 11 Objective and Constraint Functions –How do we formulate such functions? –Functions are usually not explicit –We can approximate them –Change design variables –Test different designs (sample points of test) –How many designs do we need? –How much error is there? Objective/Constraint Function x3 x2 x1

The University of SydneySlide 12 – Response surfaces are approximate objective and constraint functions based on sampled points. Create a “surrogate model”. –Useful in biomedical applications, where functions may be overly complex –Difficult to approximate functions containing singularities/discontinuities using RSM –Response is typically fitted to polynomial functions, such as this quadratic example The Response Surface Method (RSM) Coefficients to be determinedError term Design variables

The University of SydneySlide 13 –A generalised response surface, ỹ (x) in terms of basis function, φ j (x). –We can determine vector a, which describes the coefficients. –Minimum number of samples (M) should be greater than the number of unknowns or length of vector a (N). A Generalised Response Surface

The University of SydneySlide 14 Design of Experiments (DoE) –How do we know what sample points to take in the design space? –This will determine how many simulations you need to run –You can use different methods, like factorial, Koshal, composite, Latin Hypercube, D-optimal design –r n factorial design generates evenly spaced mesh of sampling points in design space –E.g. 2 design variables (n = 2), approximation order is 3 (r = 3) – At least (1+r) n designs are needed, 16 in this case x1x1 x2x2

The University of SydneySlide 15 –Obtain FEA results (y) at selected sampling points (x (i) ) –The error term ( ε i ) between ỹ (i) and y (i) for each sample point –Least squares method is the sum of the squares of each ε i, giving total error E(a). Minimise this to get reasonable values of a. Making a Response Surface RSM results FEA results

The University of SydneySlide 16 RSM in Matrix Form FEA results RS results Error

The University of SydneySlide 17 –Relative error can be calculated based on FEA results –Error comes from: –Basis function selection –Sampling point selection –Least square minimisation –Other stats techniques, like ANOVA (analysis of variance) can be used to identify effect of design variables on response RSM Error Evaluation FEA results RS results

The University of SydneySlide 18 Case Study: Functionally Graded Dental Implant Boundary condition Richer in HAP/Col y 0 Crown Implant Cortical bone Cancellous bone

The University of SydneySlide 19 Functionally Graded Material (FGM) Design Variables and Objectives m defines the material gradient and is the design variable.

The University of SydneySlide 20 FGM Response Surfaces –Polynomial basis function

The University of SydneySlide 21 Multiobjective Optimisation (Pareto Solutions) –Three objective functions – Cortical density – Cancellous density – Displacement –Weighted sum algorithm –Evolutionary algorithms – MOGA – MOPSO

The University of SydneySlide 22 Fixed P1 P2 F Workbench Optimisation Bi-objective functions: –Min σ vm,peak –Min δ s.t. –Length (P1) –Young’s modulus (P2) –10 < P1< 200 mm –50 < P2 < 200 GPa

The University of SydneySlide 23 Project Schematic showing the simulation, parameter, RSM, and optimisation modules set up Project Overview

The University of SydneySlide 24 Import geometry and assign parameter values. In this case, the implant length is varied. Geometry Parametisation

The University of SydneySlide 25 Create a material property variable in ‘Engineering Data’. Tick the parameter box next to Young’s modulus to make it a changeable parameter in our simulations. Material Parametisation

The University of SydneySlide 26 Create FE mesh, apply loads and boundary conditions. Pick ‘eqv. Stress’ and ‘total deformation’ under ‘Solution’, and run the FE analysis once. View results >> and pick the max. Eqv. Stress and the max. Total deformation as parameters by ticking the parameter boxes as shown below. These are the objectives to minimise/maximise in our design. Workbench Objective Function Definition

The University of SydneySlide 27 Add the response surface module to the project and connect with the parameter set. Then modify the parameter upper, and lower bounds. Create sample points and run simulations on all these sample points. Parameters Output Workbench Sample Point Creation

The University of SydneySlide 28 Plot 3D response surface curve following DOE completion. There will be two surfaces. One for the max. eqv. stress. The other for max. total deformation. Workbench Response Surface

The University of SydneySlide 29 Add Goal Driven Optimization module and connect with Response surface and parameter set. Change optimization parameters to MOGA (multi-objective genetic algorithm). Pick objective functions and optimize. View Pareto curve. ` Pareto in Workbench

The University of SydneySlide 30 What did we learn? –Direct or inverse approaches to design optimisation –Creation of surrogate models from finite element analysis –Defining a design optimisation framework –RSM to approximate responses/objective functions –Design of experiments –Error calculation –Multi-objective optimisation often results in trade-off –Workbench tools for optimisation