Topology Optimisation Using the Level Set Method Dr H Alicia Kim - PowerPoint PPT Presentation

Topology Optimisation Using the Level Set Method Dr H Alicia Kim Department of Mechanical Engineering

Researchers • Chris Bowen, Chris Budd, Julian Padget • Dave Betts, Peter Dunning, Yi-Zhe Song, Joao Duro • Chris Brampton, Phil Browne, Kewei Duan, Caroline Edwards, Peter Giddings, Tom Makin, Vincent Seow, Phil Williams Department of Mechanical Engineering

The Level Set Method • Front or boundary tracking method • Commonly used in image processing, moving boundary problems, multiphase problems, movies, etc … • Level set topology optimisation since 2000 (Sethian and Wiegmann), < 20,000 papers (Google Scholar) Department of Mechanical Engineering

Example: Level-Set (3D) Cantilever Beam with Vertical Load Department of Mechanical Engineering

Element-Based Method (SIMP) Cantilever Beam with Vertical Load Department of Mechanical Engineering

The Level Set Method % φ ( x ) > 0, x ∈ Ω S ' ' φ ( x ) = 0, x ∈ Γ S & ' φ ( x ) < 0, x ∉ Ω S ' ( ( ) • Update by solving discrete φ x Hamilton-Jacobi equation k V n , i k + 1 = φ i k − Δ t ∇ φ i http://en.wikipedia.org/wiki/File:Level_set_method.jpg φ i • Naturally splits and merges holes Department of Mechanical Engineering

Level Set Topology Optimisation Method 1. Define the design problem. ( ) < 0 φ x ( ) > 0 φ x 2. Finite element analysis to compute boundary shape ς sensitivities, . Ω S ς ( u ) = A ε ( u ) ε ( u ) Where A = material property, ε = strain ( ) = 0 φ x Department of Dunning and Kim (2011) FEA&D Mechanical Engineering

Level Set Topology Optimisation Method 3. Level set functions updated using a Hamilton-Jacobi equation k V n , i k + 1 = φ i k − Δ t ∇ φ i φ i where , , λ = Lagrange multiplier for a constraint and V n = λ − ς ( u ) Δ t = iterative time step. 4. λ is determined by Newton’s method. 5. Gradient, Δ φ is computed using the upwind finite difference scheme and higher order weighted essentially non-oscillatory method (WENO). 6. Check for convergence and iterate. Department of Mechanical Engineering

How does it create a new hole? Where to create a hole is not difficult, when to create is! Department of Mechanical Engineering

Previous Hole Creation Approaches Methodology Challenge Start with a random number of Does not create holes; solutions holes. dependent on the initial design. Topological derivative to create a Does not link to shape derivative hole. so optimisation of boundaries and hole creation are unrelated. Topological derivatives are Convergence can be slow. exclusively used. Holes are created at regular The selection of the interval is intervals. arbitrary and can slow the convergence. Hole creation criteria based on Heuristic, fundamentally does not stress or strain energy. link to shape derivative. Department of Mechanical Engineering

Our Approach of Creating a Hole φ • Introduce a secondary level set function, . • Describes the additional fictitious dimension, bounded by − h ≤ φ ≤ + h • is updated using the Hamilton-Jacobi equation φ k + 1 = φ i k − Δ tV n , i φ i where à establishes link between shape V n = λ − ς ( u ) and topological optimisation. • Hole creation only when more optimal than shape optimisation. Department of Mechanical Engineering Dunning and Kim (2013) IJNME

Cantilevered Beam in 2D 160 × 80, 50% volume, 106 iterations Department of Mechanical Engineering

Cantilevered Beam in 2D Department of Mechanical Engineering

Cantilevered Beam • 45% volume constraint • 0.5% difference in compliance • Robust with respect to the initial design Department of Mechanical Engineering

Robust Topology Optimisation Department of Mechanical Engineering

Robust Topology Optimisation • Minimisation of expected and variance of performance ! $ ! $ [ ] + 1 − η J = η & E C & Var [ C ] # # w 2 " w % " % n ∏ where ∫ E [ C ] = C ( u , f ) P ( f i ) df f i = 1 n ∫ C ( u , f ) 2 ∏ E [ C ] 2 Var [ C ] = P ( f i ) df − f i = 1 • Topology optimisation + Uncertainties = conventional methods are computationally intractable Department of Mechanical Engineering

We have shown: • The robust energy functional has an analytical minimum • Can be solved by a small set of auxiliary problems • Uncertainties in magnitude of loading m m – Expected compliance: (N+1) cases ∑ ∑ 2 [ ] = ∴ E C κ ij µ i µ j + κ ii σ i i , j = 1 i = 1 – Variance: (N+3) cases n ! # ∑ ( ) C = C 1 ( u , f , θ ) + w 1, i C 2, i ( u ,1, µ θ i ) + w 2, i C x , i ( u ,1, θ x ) + C y , i ( u ,1, θ y ) " $ i = i • Uncertainties in direction of loading ( f ix = f i cos θ i and f iy = f i sin θ i ) – Expected compliance: (N+1) cases m m m m m ( ) ( ) ∑ ∑ ∑ 2 ∑ ∑ 2 σ i 2 σ j 2 Var [ C ( f )] = 4 + 2 κ i . k κ j , k µ i µ j σ k κ i . j i = 1 j = 1 k = 1 i = 1 j = 1 Department of – Variance: 256 à 23, on-going. Mechanical Engineering

Example: Column under compression A single load with uncertainty in direction, 20% volume Initial design Deterministic solution Department of Mechanical Engineering

Example: Column under compression σ θ =0.3 σ θ =0.1 σ θ =0.2 E[J]: Det. sol. = 525 E[J]: Det. sol. = 1124 E[J]: Det. sol. = 2045 Robust sol. = 377 (22%) Robust sol. = 449 (60%) Robust sol. = 523 (74%) Department of Mechanical Engineering

Example: Double hook Uncertainties in magnitude: µ = 5.0, σ = 0.5 Uncertainties in direction of loading: µ = 3 π /2, σ = 0.25 50% volume Deterministic solution Robust solution E[C] = 2.36 E[C] = 1.50 Department of Mechanical Engineering

Example: Beam ! $ ! $ [ ] + 1 − η J = η 160 x 80, 40% volume & E C & Var [ C ] # # w 2 " w % " % Initial Design Deterministic Solution µ 1 =1.0 µ 2 =1.0 µ 3 =1.0 σ 1 =0.5 σ 2 =0.1 σ 3 =0.2 Department of Mechanical Engineering

Example: Beam ! $ ! $ [ ] + 1 − η J = η & E C & Var [ C ] # # w 2 w " % " % Department of Mechanical Engineering

Robust Solutions for Varying Weights ! $ ! $ [ ] + 1 − η J = η & E C & Var [ C ] # # w 2 " w % " % Department of Mechanical Engineering

Multidisciplinary Topology Optimisation Department of Mechanical Engineering

Aircraft Wing Aero-Structural Optimisation x ' Topology optimization be used to explore alternative designs • (a) - bars in main and intermediate groups of optimal design. Mostly applied to a pre-determined layout or an individual • component Eschenauer, Becker & Schumacher 1998 Balabanov & Haftka 1996 all bars of optimal design. HSCT wing with rigid fuselage. 1402 Maute & Allen 2004 Stanford, Beran & Bhatia, 2013 Department of Mechanical Engineering Stanford & Beran, 2010 For Peer Review

3D Level Set Topology Optimisation of a Wing • Objective to optimize 3D topology of wing box domain • Including aero-structural coupling is important: o Loading dependent on deformed shape of the wing o Analysis & sensitivity computation Department of Mechanical Engineering

Aerostructural Topology Optimisation • Minimize: Total structural compliance • Subject to: Lift ≥ Weight C = total compliance u = structure displacement • Aerodynamic loading from single flight condition vector f = total load vector • Fixed angle of attack K = structure stiffness matrix L = total lift ( ) = f ( u ) T u W c = fixed aircraft weight Minimize : C u W b = wing structure weight ( ) ≥ W c + W b Subject to : L u f c = fixed load vector Q = aerodynamic stiffness Ku = f ( u ) = f c + Qu matrix Department of Mechanical Engineering

3D Level Set Topology Optimisation of a Wing Aerodynamics: Doublet Lattice Method • Fluid-structure interaction: Finite Plate Spline (work conserved) • Structures: Finite Element Analysis • DLM mesh FPS mesh FEA mesh Department of Mechanical Engineering

Example: 3D Wing Box • Aspect ratio 6, Chord 2m Compliance Weight Lift Design • Wing box from leading edge to (Nm) (kN) (kN) 80% chord Initial 897.1 19.92 19.94 • Wing box depth 15% Chord Optimum 870.5 19.88 19.88 • Discretization: 32 × 120 × 6 elements Leading edge Root Initial design x-section Department of Mechanical Engineering

Example: 3D Wing Box Department of Mechanical Engineering

Composite Tow Paths Optimisation Department of Mechanical Engineering

Advanced Composite Materials Department of Discontinuous Fiber Angles. Continuous Fiber Angles. Mechanical Engineering

Composite Tow Paths Optimisation • Our approach: use the level set method • Optimise the tow paths not the fibre angles • Ensures continuity of fibre angles • Initial solution: topological optimum with isotropic material • Single and multiple level set functions • Minimise compliance Department of Mechanical Engineering

Test Models Cantilever Beam: Plate Loaded Out of Plane: Department of Mechanical Engineering

Cantilever Beam Result Initialisation Final Solution Elemental Initial Solution % FCS ¡ Compliance ¡ Compliance ¡ Difference ¡ Solution Level Set 26.94 ¡ 19.30 ¡ 18.99% ¡ 93.12% ¡ Method ¡ Elemental 34.42 ¡ 16.22 ¡ - ¡ 66.0% ¡ Method ¡ Department of Mechanical Engineering