Fictitious Domain Methods and Topology Optimization

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1 Fictitious Domain Methods and Topology Optimization Martin Berggren UMIT research lab Department of Computing Science Umeå University April 11, 2014 Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

Geometric flexibility The geometric flexibility of the FEM an important reason for its success The FEM works well with unstructured meshes Unstructured meshes can be generated fast and

2 Geometric flexibility The geometric flexibility of the FEM an important reason for its success The FEM works well with unstructured meshes Unstructured meshes can be generated fast and automatically in many cases Unstructured mesh (from Comsol Multiphysics manual) Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

3 Typical FEM procedure Three distinctive steps: Preprocessing: Geometry definition (often from CAD) + generation of body-fitted mesh Analysis: Assembly of system matrices, solution of the algebraic equations Postprocessing: Visualization, qualitative and quantitative analysis of the solution Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

4 Limitations of the typical procedure The clear separation between preprocessing and analysis sometimes problematic: The computational domain changes with time, e.g. fluid structure interaction The geometry is a part of the solution of the problem, e.g. design optimization, free boundary problems In those cases, can be an advantage to integrate geometry handling and analysis Fictitious domain methods (also called domain embedding methods) accomplish this Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

5 Fictitious domain methods Basic idea: A large (often square) computational domain covers real, physical domain The computational domain is meshed using a fixed (often regularly structured) mesh. The equations are solved on the larger, computational domain The boundary conditions on the real domain becomes interior conditions in the computational domain The point with it all: easier to change the interior conditions than to remesh and interpolate on the new mesh Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

6 Fictitious domain methods The boundary conditions at the interior boundary have to be forced somehow. Two basic ways for this: Through extra or modified boundary integrals in the variational form Through extra or modified. volume integrals in the variational form Today: a basic (but very useful!) volume integral version. Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

7 Natural vs. essential boundary conditions The three standard boundary conditions for second-order elliptic boundary-value problems: (i) Dirichlet, (ii) Neumann, and (iii) Robin conditions in (1a) Dirichlet on (1b) Neumann on (1c) Robin on (1d) Note: Case (1c) requires for consistency. Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

8 Variational forms Multiply equation (1a) with a test function, integrate, and use Green s formula: Dirichlet BC: let on Neumann BC: on Robin BC: on : the space of square-integrable functions with square-integrable derivatives Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

9 Variational forms : the subspace of functions that vanish on the boundary PDE with Dirichlet boundary condition Var. form: Essential boundary condition Find such that PDE with Neumann boundary condition Var. form: Natural boundary condition Find such that PDE with Robin boundary condition Var. form: Natural boundary condition + bdy. integral Find such that Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

10 FE approximation & implementation FEM: variational form solved in space of continuous, piecewise polynomial functions on a mesh with nodes. Expanding in a nodal basis: for Essential boundary conditions are enforced explicitly on. Standard approach for on the boundary, remove index for nodes on boundaries int int for Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

11 FE approximation & implementation In contrast to essential boundary conditions, the natural conditions coming from Neumann boundary condition are invisible in the variational form! Suggests a fictitious domain approach! Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

12 The material distribution approach : the th element in a mesh with totally elements. For Neumann boundary conditions, the variational form can be written Let. Define the element material indicator if, otherwise ( ) and extend the variational form into, meshed with elements: or, equivalently Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

13 The material distribution approach Solution undefined in ; stiffness matrix becomes singular. Cure: use modified material indicator: (, say ) if, otherwise ( ) Fixed mesh: the geometry is now defined by the values of function An image (pixel) based geometry representation Very fine mesh needed for accurate geometry representation Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

14 Material distribution optimization The element material indicator has often a natural physical interpretation: local heat conductivity, thickness in depth direction, density of material, etc. Material distribution optimization: use the values of as decision variables in optimization Material distribution optimization perhaps the premiere application of this particular fictitious domain approach! Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

15 Example 1 (linear elasticity) Half of the material is cut out to obtain a structure as stiff as possible A so-called topology optimization problem: how the body is connected (the number of holes) is not given a priori remove 50 % The problem is to divide a ground structure ( ) into a material region () and a non-material (hole) region ( ) Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

thin sheet of material of variable thickness Find the stiffest beam subject to Г c Ω ^ t for a

16 Example 2 (the variable thickness sheet) A cantilever beam (sv. konsolbalk) is clamped to a wall and vertically loaded at the other side The beam consists of a thin sheet of material of variable thickness Find the stiffest beam subject to Г c Ω ^ t for a given ρ Results for Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

17 Material distribution optimization for elasticity First numerical demonstration by Bendsøe & Kikuchi (1988). Intensely developed since then Used for advanced component design, particularly in the car and aeronautical industries Three applications of topology optimization carried out on the Airbus A380 aircraft is estimated to have contributed to weight savings in the order of 1000 kg per aircraft Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

18 Example 3 (acoustical labyrinth ) Fill the inside of a funnel-shaped device in order to maximize the transmitted sound energy. From Wadbro & Berggren (2006): Ω m Ω m Ω m Optimized for frequencies,,, Hz R Frequency Optimized for frequencies,,, Hz R Frequency Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

19 Material distribution optimization with FEM Conceptually, all three examples use a similar strategy: The material indicator function is used as decision variables in the optimization algorithm With a crude mesh, the pixel image representation of the geometry is noticeable: Design variable : represents thickness in normal direction Design variable : represents absence/presence of material Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

20 Material distribution optimization with FEM For the optimization, we introduce an objective function, a numerical performance measure. E. g. the elastic stiffness (example 1 & 2), or the acoustic transmission properties (example 3) The objective function is extremized (maximized or minimized) under constraints, e. g. We use gradient-based optimization method. Needs to calculate, where is the design value in element. For the example problems above, gradients are easy to obtain by post processing of the numerical solution Iterative procedure: guess in all elements, calculate solution, calculate derivates, update, etc Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

21 The acoustical optimization problem: details Outline: 1. Acoustics modeling 2. Material modeling 3. The optimization problem 4. Some additional results Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

22 The wave and Helmholtz equations The acoustic pressure (pressure fluctuations) satisfies the wave equation We consider time-harmonic solutions at angular frequency. The complex-valued amplitude function then satisfies that is, the Helmholtz equation where, the wave number Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

23 Plane wave solutions Functions satisfy the Helmholtz equation Plane wave traveling towards increasing : Plane wave traveling towards decreasing : Plane waves occur in straight narrow channels (acoustic wave guides) Assume now that we put a conical-shaped horn at the end of a wave guide Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

24 The acoustic horn P(x,y,t) = e iωt (Ae ikx + Be ikx ) Г in Parts of plane wave in the wave guide is reflected back, parts is transmitted to free space When size of horn, the reflection properties will be sensitive to horn shape Loudspeaker: want low reflections at all transmission frequencies Brass instrument: want the waves to be reflected with zero phase shift at certain pre-specified frequencies to build up standing waves in the feeding wave guide The horn directs the sound towards the front of the horn Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

25 Input/Output perspective Input Output P(x,y,t) = e iωt Be ikx P in (x,y,t) = e iωt Ae ikx Г in Г in p 1 (µ ) Inviscid case: 240 Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, /

26 Loudspeaker optimization objectives Transmission efficiency: Minimize, where in is the reflection coefficient. Equivalent to maximizing total exterior output Directivity: The angular distribution of sound in terms of the far-field pattern p 1 (µ ) Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

27 Computational domain Γ out Ω R Ω Γ n Γ sym Γ in Γ sym 2D Helmholtz equation, planar symmetry Symmetric up/down with respect to boundary sym The infinite domain truncated at out, located at radius. Need artificial boundary condition to absorb outgoing waves A plane wave source in waveguide, at in, imposing a right-going wave with amplitude All other boundary parts,, are sound hard. Constitute boundaries to solid materials Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

28 Boundary conditions Condition of symmetry and presence of sound-hard boundaries enforced by at sym and No pressure gradients at walls no flow of air through wall. Only plane waves in wave guide:. If in is located at, condition at in sets amplitude for the right-going wave and absorbs the left-going wave Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

29 Artificial boundary condition The so-called first-order Engquist Majda radiation condition at out absorbs reasonably well outgoing waves that travel perpendicular to out Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

30 The boundary-value problem Γ sym Γ in Γ out Ω Γ n Γ sym R Ω in (2a) on out (2b) on in (2c) on (2d) Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

31 Variational form Recall: for the FE discretization, need to write the equations in variational form: 1. Multiply the PDE with a test function and integrate over the computational domain 2. Integrate terms with second derivatives by parts, utilizing Green s identity 3. Substitute the boundary condition expressions to get rid of terms Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

32 Variational form Multiply (2a) with and integrate: Green s identity Boundary conditions out in in Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

33 Variational form Rewriting the expression yields that the solution satisfies out in out in for all test functions with square-integrable derivatives. Γ out The boundary conditions on in and out enters as boundary integrals Ω Γ n R Ω As before: the Neumann boundary conditions at sound-hard (and symmetry) boundaries do not enter in the variational form! Γ sym Γ in Γ sym In particular, the material regions inside the horn is noticed only through its absence from the integrals! Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

34 Material modeling Extend the domain to, where is the sound-hard material region inside the horn Here the material indicator signifies air or sound-hard material if (air) if (sound hard) Then the variational form can be written out in out out The function constitutes the design variables: the decision variables in the optimization algorithm Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

35 Manipulating the design variable Optimization using numerically difficult. Relax the feasible set to the interval. Allows gradient-based algorithms To force either or, penalize the objective: in : penalty parameter To avoid microstructures on the element level, define indirectly through a filter: where integral kernel has support only in a radius around point Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

36 Relaxation, penalty, filtering, sharpening Minimizing reflection at Hz (Wadbro & Berggren 2006) No penalty: ; Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

37 Relaxation, penalty, filtering, sharpening Minimizing reflection at Hz (Wadbro & Berggren 2006) No penalty: ; Penalty:. Mesh dependent microstructures appear Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

38 Relaxation, penalty, filtering, sharpening Minimizing reflection at Hz (Wadbro & Berggren 2006) No penalty: ; Penalty:. Mesh dependent microstructures appear Penalty and filter mesh independence (note the gray areas of size!) Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

39 Relaxation, penalty, filtering, sharpening Minimizing reflection at Hz (Wadbro & Berggren 2006) No penalty: ; Penalty:. Mesh dependent microstructures appear Penalty and filter mesh independence (note the gray areas of size!) Sharpening: removing the filter and continue the iterations Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

40 Example 3 revisited Simultaneous optimization over freq frequencies: freq in Optimized for frequencies,,, Hz R Frequency Optimized for frequencies,,, Hz R Frequency Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

Example 4: Interior layout of a reactive muffler l Γ s Ω m Ω d Ω p Γ L r1 r 2 l Ω Γ sym Γ I Γ R Cylindrical symmetry Perforated pipe possibly running from inlet to outlet 1 Placement of sound-hard

41 Example 4: Interior layout of a reactive muffler l Γ s Ω m Ω d Ω p Γ L r1 r 2 l Ω Γ sym Γ I Γ R Cylindrical symmetry Perforated pipe possibly running from inlet to outlet 1 Placement of sound-hard material in subject to design Reactive muffler: opposite objective to the horn: minimize the transmission through the muffler! 1 With Esubalewe Yedeg and Eddie Wadbro Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

42 The making of a Helmholtz resonator Minimizing transmission at 349 Hz, no pipe (a) (b) (c) (d) (e) with filter with filter dropping filter after 2nd post processing Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

43 Multifrequency transmission minimization 20 frequencies, Hz. Exponentially spaced (a) (a) without pipe (b) (b) with pipe Transmission Loss (db) (c) (c) Transmission loss. without pipe. with pipe Frequency (Hz) Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

44 Example 5: Interior layout of a subwoofer cm... Design variables: cm voxels air/sound hard material Maximize radiated power, fixed input voltage Hybrid model: cm 3D Helmholtz models around driver (FEM) and exterior (BEM) 2D Helmholtz in box Mechanical and electrical circuits for driver Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

45 Preliminary results Work in progress! Optimized Hz Optimized Hz SPL 1m [db] SPL 1m [db] Empty 50 Empty Optimized Optimized Frequency [Hz] Frequency [Hz] Martin Berggren (Umeå University) Fictitious Domain Methods and Topology Optimization April 11, / 42

Timo Lähivaara, Tomi Huttunen, Simo-Pekka Simonaho University of Kuopio, Department of Physics P.O.Box 1627, FI-70211, Finland

Timo Lähivaara, Tomi Huttunen, Simo-Pekka Simonaho University of Kuopio, Department of Physics P.O.Box 627, FI-72, Finland timo.lahivaara@uku.fi INTRODUCTION The modeling of the acoustic wave fields often