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Abstract - This paper describes the actual stage of the research work dedicated to the investigation, development, implementation and validation of a new-advanced shape optimization approach, particularly tailored for two-dimensional electric structures with complex geometry. The proposed numerical approach is based on the efficiency of the Extended Finite Element Method and the flexibility of the Level Set Method, to handle moving interfaces without remeshing at each optimization step. The result is a powerful and robust numerical shape optimization algorithm that demonstrates outstanding suppleness of handling topological changes, fidelity of boundary representation and a high degree of automation, in comparison with other methods reported in the literature.
The result is a powerful and robust numerical shape optimization algorithm, that demonstrates outstanding suppleness of handling topological changes, fidelity of boundary representation and a high degree of automation, in comparison with other methods reported in the literature.
Keywords: Shape Optimization, Level Set Method, Extended Finite Element Method, Genetic Algorithm, Electric Structures.
In the classical domain of structural optimization, two main techniques have been mainly studied and are now well-known, the topology and the shape optimization. These two techniques have reached a certain degree of robustness and sophistication but still present some major drawbacks , .
When topology or shape optimization using Finite Element Method (FEM) is involved, a large number of mesh adaptations or even re-meshing is required during the optimization process. After each optimization a new geometry is obtained and a new distorted mesh is needed. When, during the optimization process excessive mesh distortion occurs, the solution accuracy is influenced and a totally new mesh is to be constructed. Making robust algorithms for topology and shape optimization tools based on this approach remains a real problem i.e. book keeping, mapping recalculation, etc. , .
Recent developments in mechanical engineering based on modeling discontinuities, such as material interfaces without remeshing, provide challenging opportunities. This is due to the so called eXtended Finite Element Method (XFEM) , , ,  and .
On the other hand, very powerful mathematical techniques are available now in order to deal with moving boundary problems as the Level Set Method (LSM). Using LSM instead of performing geometrical operations, a convection equation is solved and it provides the new geometry, including topology changes , , , .
In order to avoid the main problems encountered with the classical optimization techniques, we propose in this paper a new shape optimization approach that couples the efficiency of the XFEM with the flexibility of the LSM.
This approach may present all the advantages of these two methods and avoid the main difficulties associated with them. Is based on two steps: the first one is the analysis method and the second one the representation of the geometry. The analysis is realized using the XFEM. It enables to include in the design features, such as interfaces between material and void that are not coincident with the mesh. Hence, the mesh perturbation present in the shape optimization is suppressed as we keep the same mesh during all the optimization steps. The description of the geometry is represented by the zero iso-contour of an implicit function called the Level Set function .
This new shape optimization approach based on XFEM and LSM exhibits really promising characteristics, as it allows deep topological changes and a very flexible modeling of the geometry. However numerical applications pointed out some specific difficulties and problems to be handled in the implementation.
In the next section, a brief introduction of the XFEM, applied for modeling a two-dimensional stationary electric field is illustrated. In section 3, the basis of the LSM, choice of the level function and computation of the velocity are described. In sections 4 and 5, details about the algorithm, a study case and obtained results are presented. Conclusions and perspectives of the research are given in the last section.
The extended finite element method
Discontinuities, as for instance the moving material interfaces, play an important role in many types of optimisation problems. Scientific world has given more and more attention in the last years to the problems where it is necessary to model the motion of these kinds of discontinuities . Due to the fact that standard FEM are based on piecewise differentiable polynomial approximations, they are not well suited to problems where the solutions contain discontinuities, discontinuities in the gradient, singularities or boundary layers. Typically, FEM requires significant mesh refinement or meshes that conform with these features, to yield acceptable results , , .
In response to this deficiency of standard the FEM, the extended Finite Element Method is a novel approach tailored to simulate problems involving moving discontinuities. Initially the methodology was introduced for the analysis of crack propagation . However this method has been quickly applied to several other types of problems such as elastic problems, solidification, two-phase flow, contact, composites, etc. [1-9]. An initial form of the method is reported in Belytschko and Black . The methodology has recently been generalized in Belytschko et al . The approach is based on a local partition of unity as in Chessa et al.  and Melenk .
To our knowledge, this method has not yet been applied to optimisation of electrical structures. We consider that the proposed approach based on the efficiency of the XFEM and the flexibility of the LSM for the optimal design of complex electric devices is original and innovative and provides more efficient, stable, accurate and faster solutions in comparison with any other available tool.
The basis of the Extended Finite Element Method
In contrast with the FE meshes, where the mesh conforms to the interface, the XFEM uses a fixed mesh which does not need to conform to the interface. This is done by extending the standard FE approximation with extra basis functions that capture the behaviour of the solution near the interface , . This is particularly useful for problems involving moving interfaces where the mesh would otherwise require regeneration at every time step, i.e. shape optimization problems. Consider the elliptic equation:
Embedded within ?, there is an interface ? (t) as in Figure 1. The coefficients ?, ? and f may be discontinuous across ? (t) and jump conditions are given on the interface. Inside the studied domain ?, there are two sub-domains with different material properties. This type of problem arises in a broad spectrum of mathematical models and hence, a wide range of numerical methods have been devised to solve it. Often, the location of ? (t) varies in time.
As a result, methods which are easily adapted to an arbitrary ? (t) are important. In order to reduce the continuum problem described by equation (1) to a discrete system, the XFEM approach is used . The domain ? could be meshed by an arbitrary FE mesh, but in this paper it is meshed with regular triangular elements, independent of the moving interfaces ? (t). The XFEM approximation is:
To include the interface's effect, enrichment functions are added to the standard finite element approximation for each element cut by the interface.
Three types of elements are defined:
- fully enriched elements, all the nodes of the triangle elements are enriched with the enrichment function;
- partially enriched elements, some nodes which belongs to the triangle elements are enriched with the enrichment function;
- unenriched elements, no nodes are enriched. These are the classical FEM elements.
In other words, only the elements near the material discontinuity ? (t) support extended shape functions, whereas the other elements remain unchanged. With this assumption the gradient discontinuity at the interface ? (t) is computed as:
The choice of enrichment function is based on the behaviour of the solution near the interface. These functions are chosen a priori by the knowledge of the physical problems at hand. In Table 1 are given the most typical enrichment functions.
This enrichment function yields only continuous solutions. The advantage is that it automatically satisfies the continuity condition [u] = 0 and does not require the use of Lagrange multipliers (as for example the Heaviside function).
The graphics of the discontinuous shape functions obtained by multiplying a classical Nj (x) shape function with an enrichment function are given in Figure 3. Case (a) match to a standard FE shape functions, while case (b) to a Ramp enrichment function. The discontinuity is visibly observed in Figure 3(b).
The change of the classical FE field approximation does not introduce a new form of the discretized finite element matrix equation, but leads to an enlarged problem to be solved , , :
Evaluating the domain integral terms requires a numerical quadrature method. Elements away from the interface ?(t) are evaluated using standard Gaussian quadrature. Elements that are cut by the moving interface ?(t) must be treated differently due to discontinuities in the coefficients and the enrichment function. The interface is first interpolated as a line segment (ab) and the element is divided into triangles that conform to the interface as in the figure below. In this case the integration term becomes:
The level set method
The explicit representation of the interface that is used in the classical FEM forbids deep boundary or topological changes, such as creation of holes. This limitation is the main reason of the low performance generally associated to the shape optimization problem. In opposite, the LSM developed by Osher and Sethian, which consist of representing the interface with an implicit function, overcomes this kind of deep changes , , .
The basis of the Level Set Method
The LSM is a numerical technique first developed for tracking moving interfaces. It is based on the idea of representing implicitly the interfaces as a level set curve of a higher dimension function.
The moving interface , considered between the two sub-domains ?1 and ?2 with different conductivity (see Figure 1), is conventionally represented by a zero Level Function, and the sign convention, as follows :
Appling the XFEM framework, the Level Set is defined on the structured mesh and at each finite element node is associated a geometrical degree of freedom representing its Level Set function value. The Level Function is discretized on the whole design domain ?, with the standard FE and in all cases the same mesh and shape functions are used as for the dependent variable :
numerical approach application
The test problem consists of a rather academic case, as given in Figure 8 where a 2D cross-section of a resistor pattern is given.
Inside the studied domain ?, there are two sub-domains with different conductivity, respectively ?1 with ?1 and ?2 with ?2. Denote the union of the interfaces between ?1 and ?2 by ?(t). The resistor terminals are marked with thick lines and are considered having a rigid position.
The resistance is computed using a stationary electric field model, without charge distributions inside ?2 and governed by a Laplace's equation :
where u represents the electric potential distribution, and ? the electric conductivity. The boundary conditions attached to equation (16) are of Dirichlet type on the terminals 1 and 2 (u = constant) and of Neumann type on the insulated boundary. The potentials of these terminals are: u1 = 10 (V) and u2 = 0 (V), respectively. The conductivity of the resistor (sub-domain ?2) is and 1 mm is the thickness of the resistor. All the dimensions of the electric structure are given in mm (Figure 9). The used structured finite element mesh in the XFEM approach, non-conform to the interface is given in the figure below.
The optimization objective is to minimize the value of the resistor, equivalent to find the shape of the interface ?(t) which assures this value. In this work, the evolution equation of the Level Set is used (Equation 15) and the nodal Level Set values are the design variables. The optimization procedure was realized using a genetic algorithm, in order to avoid the sensitivity analysis computation .
The applied methodology should be synthesised as below and consists in the following steps:
- Problem definition
- Optimization definition (Objective function, ending criteria, GA parameters).
- FEM Mesh generation, nonconforming to the interface.
- Solving the PDE equations using XFEM approach.
- Solving the LSM equations.
- Speed computation of the level set function.
- Re-computation of the interface according to the information obtained from LSM.
- Objective function computation.
- If the ending criteria is reached stop. If not go to step 5.
Conclusions and perspectives
In this paper, a new method based on XFEM and LSM methods for shape optimization of 2D electric structures is presented. No significant problems have been encountered in the optimization procedure and quite good results were obtained in comparison with other results found in the literature, for the similar study case , . The XFEM method has proven to be very useful: no remeshing process is needed in this application during the optimization process.
The ongoing work will extend the results already in hand: first, by computing the sensitivity analysis in order to use deterministic optimization methods, more faster as the stochastic ones; the next step will certainly focus on the development of a "real" Level Set framework in order to allow much more performance in the sense of geometrical modification.
Obviously, the coupling of the XFEM with the Level Set has proven to be a really promising method for the shape as well as for the topology optimization.
The work was supported by the National Council of Research in the Higher Education (CNCSIS) under research support scheme of the "IDEI" program, grant ID_2538/2008, "Development of a Mathematical Analysis Technique for Modeling Electrode Shape Changes in Electrochemical Processes, a New Virtual Design Tool".
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