Abstract – This paper describes the existent phase of the research work dedicated to the probe, development, execution and proof of a new-advanced form optimisation attack, peculiarly tailored for planar electric constructions with complex geometry. The proposed numerical attack is based on the efficiency of the Extended Finite Element Method and the flexibleness of the Level Set Method, to manage traveling interfaces without remeshing at each optimisation measure. The consequence is a powerful and robust numerical form optimisation algorithm that demonstrates outstanding lissomeness of managing topological alterations, fidelity of boundary representation and a high grade of mechanization, in comparing with other methods reported in the literature.

The consequence is a powerful and robust numerical form optimisation algorithm, that demonstrates outstanding lissomeness of managing topological alterations, fidelity of boundary representation and a high grade of mechanization, in comparing with other methods reported in the literature.

Keywords: Shape Optimization, Level Set Method, Extended Finite Element Method, Genetic Algorithm, Electric Structures.

### Introduction

In the classical sphere of structural optimisation, two chief techniques have been chiefly studied and are now well-known, the topology and the form optimisation. These two techniques have reached a certain grade of hardiness and edification but still show some major drawbacks [ 2 ] , [ 3 ] .

When topology or form optimisation utilizing Finite Element Method ( FEM ) is involved, a big figure of mesh versions or even re-meshing is required during the optimisation procedure. After each optimisation a new geometry is obtained and a new deformed mesh is needed. When, during the optimisation procedure inordinate mesh deformation occurs, the solution truth is influenced and a wholly new mesh is to be constructed. Making robust algorithms for topology and form optimisation tools based on this attack remains a existent job i.e. book maintaining, mapping recalculation, etc. [ 3 ] , [ 13 ] .

Recent developments in mechanical technology based on patterning discontinuities, such as stuff interfaces without remeshing, supply disputing chances. This is due to the so called eXtended Finite Element Method ( XFEM ) [ 4 ] , [ 5 ] , [ 6 ] , [ 7 ] and [ 13 ] .

On the other manus, really powerful mathematical techniques are available now in order to cover with traveling boundary jobs as the Level Set Method ( LSM ) . Using LSM alternatively of executing geometrical operations, a convection equation is solved and it provides the new geometry, including topology alterations [ 8 ] , [ 9 ] , [ 10 ] , [ 13 ] .

In order to avoid the chief jobs encountered with the classical optimisation techniques, we propose in this paper a new form optimisation attack that couples the efficiency of the XFEM with the flexibleness of the LSM.

This attack may show all the advantages of these two methods and avoid the chief troubles associated with them. Is based on two stairss: the first 1 is the analysis method and the 2nd one the representation of the geometry. The analysis is realized utilizing the XFEM. It enables to include in the design characteristics, such as interfaces between stuff and nothingness that are non coinciding with the mesh. Hence, the mesh disturbance nowadays in the form optimisation is suppressed as we keep the same mesh during all the optimisation stairss. The description of the geometry is represented by the zero iso-contour of an inexplicit map called the Level Set map [ 13 ] .

This new form optimisation attack based on XFEM and LSM exhibits truly assuring features, as it allows deep topological alterations and a really flexible mold of the geometry. However numerical applications pointed out some specific troubles and jobs to be handled in the execution.

In the following subdivision, a brief debut of the XFEM, applied for patterning a planar stationary electric field is illustrated. In subdivision 3, the footing of the LSM, pick of the degree map and calculation of the speed are described. In subdivisions 4 and 5, inside informations about the algorithm, a survey instance and obtained consequences are presented. Decisions and positions of the research are given in the last subdivision.

### The drawn-out finite component method

Discontinuities, as for case the traveling stuff interfaces, play an of import function in many types of optimization jobs. Scientific universe has given more and more attending in the last old ages to the jobs where it is necessary to pattern the gesture of these sorts of discontinuities [ 13 ] . Due to the fact that standard FEM are based on piecewise differentiable multinomial estimates, they are non good suited to jobs where the solutions contain discontinuities, discontinuities in the gradient, uniquenesss or boundary beds. Typically, FEM requires important mesh polish or meshes that conform with these characteristics, to give acceptable consequences [ 1 ] , [ 2 ] , [ 13 ] .

In response to this lack of standard the FEM, the drawn-out Finite Element Method is a fresh attack tailored to imitate jobs affecting traveling discontinuities. Initially the methodological analysis was introduced for the analysis of cleft extension [ 15 ] . However this method has been rapidly applied to several other types of jobs such as elastic jobs, hardening, two-phase flow, contact, complexs, etc. [ 1-9 ] . An initial signifier of the method is reported in Belytschko and Black [ 3 ] . The methodological analysis has late been generalized in Belytschko et al [ 5 ] . The attack is based on a local divider of integrity as in Chessa et Al. [ 13 ] and Melenk [ ] .

To our cognition, this method has non yet been applied to optimization of electrical constructions. We consider that the proposed attack based on the efficiency of the XFEM and the flexibleness of the LSM for the optimum design of complex electric devices is original and advanced and provides more efficient, stable, accurate and faster solutions in comparing with any other available tool.

### The footing 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 non necessitate to conform to the interface. This is done by widening the standard FE estimate with excess footing maps that capture the behavior of the solution near the interface [ 2 ] , [ 3 ] . This is peculiarly utile for jobs affecting traveling interfaces where the mesh would otherwise necessitate regeneration at every clip measure, i.e. form optimisation jobs. See the elliptic equation:

Embedded within? , there is an interface? ( T ) as in Figure 1. The coefficients? , ? and degree Fahrenheit may be discontinuous across? ( T ) and jump conditions are given on the interface. Inside the studied sphere? , there are two sub-domains with different stuff belongingss. This type of job arises in a wide spectrum of mathematical theoretical accounts and hence, a broad scope of numerical methods have been devised to work out it. Often, the location of? ( T ) varies in clip.

As a consequence, methods which are easy adapted to an arbitrary? ( T ) are of import. In order to cut down the continuum job described by equation ( 1 ) to a distinct system, the XFEM attack is used [ 3 ] . The sphere? could be meshed by an arbitrary FE mesh, but in this paper it is meshed with regular triangular elements, independent of the traveling interfaces? ( T ) . The XFEM estimate is:

To include the interface ‘s consequence, enrichment maps are added to the standard finite component estimate for each component cut by the interface.

Three types of elements are defined:

- to the full enriched elements, all the nodes of the trigon elements are enriched with the enrichment map ;
- partly enriched elements, some nodes which belongs to the trigon elements are enriched with the enrichment map ;
- unenriched elements, no nodes are enriched. These are the classical FEM elements.

In other words, merely the elements near the stuff discontinuity? ( T ) support extended form maps, whereas the other elements remain unchanged. With this premise the gradient discontinuity at the interface? ( T ) is computed as:

The pick of enrichment map is based on the behavior of the solution near the interface. These maps are chosen a priori by the cognition of the physical jobs at manus. In Table 1 are given the most typical enrichment maps.

This enrichment map outputs merely uninterrupted solutions. The advantage is that it automatically satisfies the continuity status [ u ] = 0 and does non necessitate the usage of Lagrange multipliers ( as for illustration the Heaviside map ) .

The artworks of the discontinuous form maps obtained by multiplying a classical Nj ( x ) form map with an enrichment map are given in Figure 3. Case ( a ) lucifer to a standard FE form maps, while instance ( B ) to a Ramp enrichment map. The discontinuity is visibly observed in Figure 3 ( B ) .

The alteration of the classical FE field estimate does non present a new signifier of the discretized finite component matrix equation, but leads to an enlarged job to be solved [ 1 ] , [ 5 ] , [ 8 ] :

Measuring the sphere built-in footings requires a numerical quadrature method. Elementss off from the interface? ( T ) are evaluated utilizing standard Gaussian quadrature. Elementss that are cut by the traveling interface? ( T ) must be treated otherwise due to discontinuities in the coefficients and the enrichment map. The interface is foremost interpolated as a line section ( Bachelor of Arts ) and the component is divided into trigons that conform to the interface as in the figure below. In this instance the integrating term becomes:

### The degree set method

The expressed representation of the interface that is used in the classical FEM forbids deep boundary or topological alterations, such as creative activity of holes. This restriction is the chief ground of the low public presentation by and large associated to the form optimisation job. In opposite, the LSM developed by Osher and Sethian, which consist of stand foring the interface with an inexplicit map, overcomes this sort of deep alterations [ 8 ] , [ 9 ] , [ 10 ] .

### The footing of the Level Set Method

The LSM is a numerical technique foremost developed for tracking traveling interfaces. It is based on the thought of stand foring implicitly the interfaces as a degree set curve of a higher dimension map.

The traveling interface, considered between the two sub-domains? 1 and? 2 with different conduction ( see Figure 1 ) , is conventionally represented by a zero Level Function, and the mark convention, as follows [ 9 ] :

Appling the XFEM model, the Level Set is defined on the structured mesh and at each finite component node is associated a geometrical grade of freedom stand foring its Level Set map value. The Level Function is discretized on the whole design sphere? , with the standard FE and in all instances the same mesh and form maps are used as for the dependant variable [ 10 ] :

### numerical attack application

The trial job consists of a instead academic instance, as given in Figure 8 where a 2D cross-section of a resistance form is given.

Inside the studied sphere? , there are two sub-domains with different conduction, severally? 1 with? 1 and? 2 with? 2. Denote the brotherhood of the interfaces between? 1 and? 2 by? ( T ) . The resistance terminuss are marked with thick lines and are considered holding a stiff place.

The opposition is computed utilizing a stationary electric field theoretical account, without charge distributions inside? 2 and governed by a Laplace ‘s equation [ 10 ] :

where u represents the electric possible distribution, and? the electric conduction. The boundary conditions attached to equation ( 16 ) are of Dirichlet type on the terminuss 1 and 2 ( u = invariable ) and of Neumann type on the insulated boundary. The potencies of these terminuss are: u1 = 10 ( V ) and u2 = 0 ( V ) , severally. The conduction of the resistance ( sub-domain? 2 ) is and 1 millimeter is the thickness of the resistance. All the dimensions of the electric construction are given in millimeter ( Figure 9 ) . The used structured finite component mesh in the XFEM attack, non-conform to the interface is given in the figure below.

The optimisation aim is to minimise the value of the resistance, tantamount to happen the form of the interface? ( T ) which assures this value. In this work, the development equation of the Level Set is used ( Equation 15 ) and the nodal Level Set values are the design variables. The optimisation process was realized utilizing a familial algorithm, in order to avoid the sensitiveness analysis calculation [ 1 ] .

The applied methodological analysis should be synthesised as below and consists in the undermentioned stairss:

- Start
- Problem definition
- Optimization definition ( Objective map, stoping standards, GA parametric quantities ) .
- FEM Mesh coevals, nonconforming to the interface.
- Solving the PDE equations utilizing XFEM attack.
- Solving the LSM equations.
- Speed calculation of the degree set map.
- Re-computation of the interface harmonizing to the information obtained from LSM.
- Objective map calculation.
- If the stoping standards is reached stop. If non travel to step 5.
- End

### Decisions and positions

In this paper, a new method based on XFEM and LSM methods for form optimisation of 2D electric constructions is presented. No important jobs have been encountered in the optimisation process and rather good consequences were obtained in comparing with other consequences found in the literature, for the similar survey instance [ 11 ] , [ 12 ] . The XFEM method has proven to be really utile: no remeshing procedure is needed in this application during the optimisation procedure.

The on-going work will widen the consequences already in manus: foremost, by calculating the sensitiveness analysis in order to utilize deterministic optimisation methods, more faster as the stochastic 1s ; the following measure will surely concentrate on the development of a “ existent ” Degree Set model in order to let much more public presentation in the sense of geometrical alteration.

Obviously, the yoke of the XFEM with the Level Set has proven to be a truly promising method for the form every bit good as for the topology optimisation.

### Recognitions

The work was supported by the National Council of Research in the Higher Education ( CNCSIS ) under research support strategy of the “ IDEI ” plan, 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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