Simulation Driven Biomedical Optimisation Lecture 5 Andrian Sue AMME4981/9981 Week 5 Semester 1, 2016 The University of Sydney Slide 1 Modelling Workflow for Biomedical Implant Verification Data acquisition Solid modelling Optimisation The University of Sydney Finite element analysis Slide 2

Optimisation Framework and Methodology The University of Sydney Slide 3 Overview How to approach an optimisation problem in implant design? Framework for optimisation Response Surface Method Example: Functionally Graded Dental Implant The University of Sydney Slide 4 Direct and Inverse Approaches Direct Analysis problem Try many different designs Make final decision

based on chosen designs The University of Sydney Inverse Design problem Find relationship between design and result Determine design from an optimal result Slide 5 Types of Structural Optimisation Parametric Topology The University of Sydney Defining geometry and/or material Discrete design variables Locus of solutions dependent on variable

Material redistribution Discretisation of structure (homogenisation) Solutions dependent on constraints only Slide 6 A Parametric Approach to Implant Optimisation Geometry Material properties Boundary conditions Loads trial and error validation What does FEM need? How can we

improve the Geometry Material Propertiesdesign? The University of Sydney What does FEM provide ? Stress Strain Displacement Stiffness Dynamic response What is expecte d from the Min stress design?Max durability Min cost Min stress shielding

Slide 7 Direct Parametric Optimisation of a Total Hip Replacement y F Ball R=15 Rod r=7.5 60 50 40 50 Change the design variables Geometry (Length) Material (Youngs Modulus) By FEA, determine biomechanical responses E L Taper 1:25 Peak von Mises stress in implant 50 50

x 1200 Interfacial stress (at bone-implant interface) Maximum deformation (mm) 20 Peak von Mises stress (MPa)... Displacement 1000 800 600 400 200 0 Ti6Al4V CoCr Composite

25 Ti6Al4V 20 CoCr 15 Composite 10 5 0 Ti6Al4V The University of Sydney Slide 8 A Framework for Optimisation Optimal objective function The objective function drives the optimisation Min y0(x), where y0(x) is the objective function

Design constraints Limiting conditions yj(x) 0, where j = 1,2,,nc Design variable vector Selected design variables x = [ x1 , x2 , , xN ]T Design space Available values of the design variable xl x xw where x is a design variable The University of Sydney Slide 9 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 The University of Sydney Youngs T x= x1 x2 ... xN = d gx gy E x4=E x3=gy x2=gx x1=d Slide 10 T

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? The University of Sydney x2 x1 x3 Objective/Constraint Function Slide 11

The Response Surface Method (RSM) 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 Coefficients to be determined Error term i =a0 + ai xij + ajk xij xik + ei y j j ( i =1, 2, , P ) k Design variables The University of Sydney

Slide 12 A Generalised Response Surface A generalised response surface, (x) in terms of basis function, 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). The University of Sydney Slide 13 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 rn 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) x1 At least (1+r)n designs are needed, 16 in this case x2 The University of Sydney Slide 14 Making a Response Surface Obtain FEA results (y) at selected sampling points (x(i)) RSM results The error term (i) between (i) and y(i) for each sample point FEA results The University of Sydney Least squares method is the sum of the squares of each i,

giving total error E(a). Minimise this to get reasonable values of a. Slide 15 RSM in Matrix Form FEA results RS results The University of Sydney Error Slide 16 RSM Error Evaluation 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 The University of Sydney RS results FEA results Slide 17 Case Study: Functionally Graded Dental Implant Crown Cancellous bone y Richer in HAP/Col 0 Cortical

bone The University of Sydney Implant Boundary condition Slide 18 107 92 77 0.33 m=10 m=5 m=1 m=0.5 m=10 m=5 m=1 m=0.5 m=0.1 0.325 Poisson's ratio

Young's Modulus (GPa) Functionally Graded Material (FGM) Design Variables and Objectives m=0.1 62 47 32 0.32 0.315 0.31 17 0.305 2 0.3 0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10

Position along implant (mm) Position along implant (mm) m defines the material gradient and is the design variable Cancellous bone density titanium m=10 m=8 m=5 m=2 m=1 m=0.5 m=0.2 m=0.1 1.10 1.05 1.00 0.95 0.90 0.85 0 4 8

12 16 20 24 28 32 36 40 44 48 number of month bone density (g/cm^3) bone density(g/cm^3) 1.15 Cortical neck density 1.94 1.92 1.90 1.88 1.86 1.84 1.82 1.80 1.78 1.76 1.74 titanium m=10 m=8 m=5

m=2 m=1 m=0.5 m=0.2 m=0.1 0 4 8 12 16 20 24 28 32 36 40 44 48 number of month The University of Sydney Slide 19 FGM Response Surfaces Polynomial basis function cancellous density function f2 1.95 1.18 1.94

1.16 density (g/cm^3) Density (g/cm^3) cortical density function f1 f1 Poly. (f1) 1.93 1.92 1.91 1.9 1.89 1.14 1.12 1.10 f2 Poly. (f2) 1.08 1.06

1.04 1.88 0 2 4 6 material gradient (m) The University of Sydney 8 10 0 2 4 6

8 10 material gradient (m) Slide 20 Multiobjective Optimisation (Pareto Solutions) Three objective functions Cortical density Cancellous density Displacement Weighted sum algorithm Evolutionary algorithms MOGA MOPSO The University of Sydney 1 min f2 =

DCancellous ( m) min f3 =u( m) s.t. 0.1 m10 1 min f = 1 DCortical ( m) min f3 =u( m) s.t. 0.1 m10 Slide 21 Workbench Optimisation F Bi-objective functions:

Min vm,peak Min s.t. Length (P1) P2 P1 Youngs modulus (P2) 10 < P1< 200 mm 50 < P2 < 200 GPa Fixed The University of Sydney Slide 22 Project Overview Project Schematic showing the simulation, parameter, RSM, and optimisation modules set up The University of Sydney Slide 23

Geometry Parametisation Import geometry and assign parameter values. In this case, the implant length is varied. The University of Sydney Slide 24 Material Parametisation Create a material property variable in Engineering Data. Tick the parameter box next to Youngs modulus to make it a changeable parameter in our simulations. The University of Sydney Slide 25 Workbench Objective Function Definition 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. The University of Sydney Slide 26 Workbench Sample Point Creation 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 The University of Sydney Output Slide 27

Workbench Response Surface 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. The University of Sydney Slide 28 Pareto in Workbench Add Goal Driven Optimization module and connect with Response surface and parameter set. Change optimization parameters to MOGA (multiobjective genetic algorithm). Pick objective functions and optimize. View Pareto curve. ` The University of Sydney Slide 29 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 tradeoff Workbench tools for optimisation The University of Sydney Slide 30