This paper presents a topology optimisation approach that combines an adjoint-based sensitivity analysis [1] with level set methods (LSM) [2] for front propagation, and the discontinuous Galerkin (DG) symmetric interior penalty (SIP) method [3]. The problems considered in this paper will be limited to the minimum compliance design of two-dimensional linear elastic structures

Adams, T.

Coombs, W.M.

Giani, S.

English

Durham Research Online

Proceedings of the 24th UK Conference of theAssociation for Computational Mechanics in Engineering31 March - 01 April 2016, Cardi  University, Cardi Topology optimisation using level set methods and the discontinuousGalerkin method*Thomas Adams, Stefano Giani and William M. CoombsSchool of Engineering and Computing Sciences, University of Durham, South Road, Durham, DH1 3LE, UK.*thomas.d.adams@durham.ac.ukABSTRACTThis paper presents a topology optimisation approach that combines an adjoint-based sensitivity analysis [1] withlevel set methods (LSM) [2] for front propagation, and the discontinuous Galerkin (DG) symmetric interior penalty(SIP) method [3]. The problems considered in this paper will be limited to the minimum compliance design oftwo-dimensional linear elastic structures.Key Words: topology optimisation; level set methods; discontinuous Galerkin method; symmetric interior penaltymethod.1. IntroductionOne of the most challenging aspects of structural design is finding the optimal layout, or topology of astructure. Topology optimisation is the most general form of structural optimisation and is concernedwith finding the boundary of a given problem domain which is optimal in that it minimises an objectivefunctional while satisfying given constraints.In this paper a topology optimisation is implemented by performing an adjoint-based sensitivity analysisto compute the gradient of the objective functional as described in [1]. The level set function (LSF) whichdefines the internal boundaries of the problem can then be advected along the steepest descent gradienttowards an optima. The LSM developed by Osher and Sethian [2] is a simple and e cient method forcomputing the evolution of a moving interface. Since it was first used for structural optimisation in [4],the LSM has been used for boundary tracking in many shape optimisation implementations and hasproven e cacy.In this paper, the spatial domain of both the physical problem and the level set transport problem arediscretised using the DGSIP method [3]. The DG method di ers from the continuous Galerkin (CG)method in that it works over a trial space of functions which are only piecewise continuous. One of themain advantages of DG finite elements is that they are trivially parallelisable. As such this work is a firststep towards exploiting parallel computing techniques to create an e cient shape optimisation methodfor large-scale problems.Following this introduction, Section 2 will present and explain an algorithm for finding the solution tothe minimum compliance problem for a cantilever beam. This will be followed by some numerical resultsfor the given problem and conclusions in Section 3.2. Topology optimisation algorithm and problem formulationFor the general case the proposed optimisation algorithm is as follows:1. Define the problem domain.2. Initialise the LSF as a signed distance function (SDF).3. Until the value of the objective functional converges:(a) compute the current state and the adjoint state through the solution of the physical and adjointproblems;6(b) compute the advection velocity; and(c) evolve the interface through the solution of the transport Hamilton-Jacobi equation.i. Reinitialise the LSF to a SDF through a geometric redistancing method.The algorithm will now be explained through the use of an example optimisation problem. The objectiveof the optimisation process will be to determine the boundary, which for a given domain, load distributionand boundary conditions, will minimise the compliance of the system whilst satisfying a given weightconstraint. Mathematically this can be stated as:inf J (⌦) where: J (⌦) =Z@⌦Ng · u ds + `Z⌦dx (1)Where J is the objective functional, ⌦ is the domain, @⌦N is the Neumann part of the boundary, g arethe surface tractions, u is the displacement field, and ` is a penalty term enforcing the weight constraint.2.1. Step 1: Defining the problem domainThe problem to be discussed in this paper aims to find the optimal topology of a 2D cantilever beamproblem, the dimensions and the boundary/loading conditions are shown in Figure 1(a). The domain isdiscretised into a mesh of 100 by 50 first order, piecewise continuous, square elements. The domain hastwo sets of elastic properties which represent the material section and the void section. The material partof the domain has Young’s modulus, E = 1Pa and Poisson’s ratio, ⌫ = 0.3, and the void is modelled as amuch more compliant material section which has a Young’s modulus E = 10 3Pa and Poisson’s Ratio,⌫ = 0.3. The value of Young’s modulus varies smoothly across the interface between the two materialsby applying a smoothed sign function to the LSF and using this as the input to a linear function whichvaries between the two values of Young’s Modulus as the sign function varies between -1 and 1.(a) Cantilever beam problem (b) Initial zero level setFigure 1: Initialisation of the problem2.2. Step 2: Initialisation of the LSF as an SDFThe interface between the material portion and the void portion of the domain is defined as the zero levelset of the LSF,  . Where the level set function is greater than zero the domain is filled with material andwhere the level set function is less than zero the domain is filled with the ersatz material which representsthe void as further described in Section 2.1.There are two main reasons for initialising the LSF as an SDF. First of all, numerical inaccuracies willoccur during the transport process if there are large variations in the gradient of the LSF, especially in theregion close to the zero level set. Secondly, initialising the LSF as an SDF and maintaining this propertythrough frequent reinitialisation greatly simplifies the level set transport equation as discussed in Section2.5.When choosing the initial LSF, one must be sure to ensure that there are su cient holes in the domain.This is because for a two dimensional problem being evolved with a time stepping scheme which satisfies astrict CFL condition, no holes can be created during the optimisation process [1]. The initial configurationcan be seen in Figure 1(b), where black represents the material and white represents the voids.72.3. Step 3(a): Computation of the state and the adjoint state through DG solution of the physical modelIn order to compute the gradient of the objective functional, an adjoint-based sensitivity analysis isperformed. The minimum compliance problem is self-adjoint and as such is simpler than the generalcase in which an adjoint state would also have to be computed at each iteration. The author points thereader towards [1] for the details of deriving the topological derivative and simply states the result thatfor the objective functional stated in (1) the topological derivative yields a vector field, V :V = bn with b =  (u) · "(u)   ` (2)In order to compute this advection velocity field, it is necessary to first compute the displacement fieldfor the current state of the cantilever beam problem. For a linear elastic stress analysis problem, the DGapproximation for the displacement field, u, is given by:Z⌦f vd⌦ =Z⌦vB · DBu d⌦  Z [[v]] · D{{Bu}} d   Z {{Bv}} · D[[u]] d  +Z µ[[v]] · [[u]] d  (3)Where   denotes the element face, f are the body forces, v is the test function, B is the strain-displacementmatrix, D is the elastic properties matrix, µ is a penalty parameter, [[q]] denotes the jump of q and {{q}}denotes the average of q. For this problem, the boundary conditions are imposed strongly. Once again theauthor directs the reader to [3], for the details on the discretisation of elliptic problems using the DGSIPmethod.2.4. Step 3(b): Computation of the advection velocityThe advection occurs only in the normal direction to the interface and so the advection velocity at eachnode is given by, b, defined in (2). The stress,  , and strain, ", can be recovered from the displacementsolution of the linear elasticity problem described in Section 2.3. The ` term is a penalty term whichdescribes the volume ratio, i.e. the ratio of material to void, at convergence. It is prescribed during theinitial iteration, for the numerical example in this paper, ` = 150.2.5. Step 3(c): Evolution of the interfaceThe evolution of the LSF occurs through the solution of a Hamilton-Jacobi equation. However, asmentioned in Section 2.2, the maintenance of the LSF as an SDF ensures that the Euclidean norm ofthe spatial derivatives is always equal to unity and as such the level set transport equation can be greatlysimplified as described in [5], which can be stated as follows:@ @t= b (4)Equation (4) is solved explicitly in time using a forward Euler method. The time step is constrained bythe CFL condition and is inversely dependent on the maximum absolute value of the advection velocityfield at each iteration and the size of the elements in the mesh. (4) is solved in space using DG finiteelements once again. For this particular equation, both CG and DG formulations are the same, however,the DG formulation of (4) can be solved independently for each element, allowing for a much moree cient solution.2.6. Step 3(c)i: Reinitialisation of the level set functionMaintaining the signed distance property of the LSF can be achieved through multiple methods. Onemethod is known as geometric redistancing (or the brute force method). For this Yamasaki et al. [5]presented a method which involves discretising the zero level set and then computing the minimumdistance from each node to the discretised interface and multiplying this by the sign of the level set atthat node. When using a discontinuous LSF the multiplication by the sign can cause discontinuities atthe interface which results in the LSF not being reinitialised as an SDF. To remedy this a smoothing isapplied to the LSF prior to the reinitialisation whereby the value of the LSF at each node is taken to bethe average of the values at each of the degrees of freedom at that node.83. Numerical results and conclusionsThe problem presented in the previous section was ran for 50 iterations. Figure 2(a) shows a smoothconvergence of the objective functional until it reaches a minimum. The final shape after 50 iterations isshown in Figure 2(b).(a) Convergence of the objective functional (b) Final topology after 50 iterationsFigure 2: Numerical resultsTo conclude, a topology optimisation algorithm using level set methods and the discontinuous Galerkinmethod has been presented and explained. The algorithm was implemented to find the optimal topologyof a 2D linear elastic cantilever beam. The results show that the algorithm has good convergence andstability properties, and the optimal design shown is similar to that published in the literature.AcknowledgementsThe first author acknowledges financial support during the completion of this work by the UK Engineeringand Physical Sciences Research (EPSRC) grant.References[1] G. Allaire, F. Jouve, and A. Toader. Structural optimization using sensitivity analysis and a level-setmethod. J. Comput. Phys., 194(1):363–393, 2004.[2] J. A. Sethian. Level set methods and fast marching methods: evolving interfaces in computationalgeometry, fluid mechanics, computer vision, and materials science, volume 3. Cambridge universitypress, 1999.[3] D. N. Arnold, F. Brezzi, B. Cockburn, and L. Donatella Marini, Unified analysis of discon-tinuous galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39(5):1749–1779,2002, 2011.[4] J. A. Sethian and Andreas Wiegmann, Structural Boundary Design via Level Set and ImmersedInterface Methods. J. Comput. Phys., 163, 489–528, 2000.[5] S. Yamasaki, S. Nishiwaki, T. Yamada, K. Izui and M. Yoshimura, A structural optimizationmethod based on the level-set method using a new geometry-based reinitialization scheme, Int. J.Numer. Meth. Eng., 83(12), 1580–1624, 2010.9

Topology optimisation using level set methods and the discontinuous Galerkin method

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Topology optimisation using level set methods and the discontinuous Galerkin method

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