Finite elements, genetic algorithms and β -splines: a combined technique for shape optimization

Finite Elements in Analysis and Design 33 (1999) 125}141

Finite elements, genetic algorithms and b-splines: a combined technique for shape optimization W. Annicchiarico, M. Cerrolaza* Faculty of Engineering, Bioengineering Center, Central University of Venezuela, P.O. Box 50361, Caracas 1050-A, Venezuela

Abstract Shape optimization consist in changing the boundary shape of the structure to "nd the optimal design by verifying the prescribed constraints. Generally, this problem has been solved by replacing the originally design problem by a sequence of explicit subproblems. They are constructed by means of local approximation concepts based on function values and derivatives at the current design point. The aim of this paper is to propose a di!erent approach to solve bidimensional shape optimization problems by using Genetic Algorithms. Boundary b-splines curves with an automatic mesh generation are used to obtain the analytical model and "nally, the "nite element method is used to estimate the "tness function. The versatility and #exibility of the proposed approach is tested and discussed in two numerical examples, showing that the technique is able to deal with real engineering problems.  1999 Elsevier Science B.V. All rights reserved. Keywords: Genetic algorithms; b-splines; Optimization; Finite elements

1. Introduction Structural shape optimization has attracted great attention from the scienti"c community. Generally, shape optimization problems consist in varying some boundaries of the model to be designed in order to improve its mechanical behavior, as for example, to reduce high stress concentrations which normally occur at corners locations or close to them. This process is usually done by imposing restrictions and by using the selected optimization method. The problem can be divided into three main tasks. The "rst step is to de"ne the geometric and the analytical models. The geometric model is where the design variables are easily imposed and it allows an explicit integration with other design tools, such as CAD or CAM systems. On the other

* Corresponding author. E-mail address: [email protected] (W. Annicchiarico), [email protected] (M. Cerrolaza) 0168-874X/99/$ - see front matter  1999 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 8 7 4 X ( 9 9 ) 0 0 0 3 0 - X

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hand, the analytical model is used to obtain the structural response of the piece, subjected to external actions. Then, a sensitivity analysis must be done to get a solution of the problem; and "nally, an appropriate optimization algorithm must be selected to solve the optimization problem in an e!ective and reliable way. Initially, many authors such as Zienkiewicz and Campbell [1], Ramakrishnan and Francavilla [2] among others, did not use geometric modeling in the shape optimization problems addressed by them. Instead, they de"ned the nodal coordinates of the discrete "nite element model as design variables. This approach requires a large number of design variables and tends to produce jagged edges shapes. In order to overcome this problem a large number of constraints must be added, which complicates the design task. Moreover, the lack of an associated geometric model does not allow the integration with powerful design tools like CAD or CAM systems. Several authors [3}6] have used mesh parametrization methods to de"ne geometric and analytical models without the complications described above. In this approach a set of keypoints or master nodes are used to de"ne the geometry entities of the mesh, e.g. lines, surfaces, volumes. Then parametric mappings are employed to map these geometric entities to the "nite element nodes. So, the shape design variables are a set of parameters used to de"ne the position of the keypoints or master nodes. However, parametrization methods are di$cult to use because the designer has to de"ne the model in terms of master nodes rather than dimensions, which is not an easy task when dealing with complex models, and they also su!er from possible mesh degradation for large design changes, since topology of the mesh is "xed as it can be observed in the work of Olho! et al. [7]. Another approach used by Belengundu and Rajan [8] is the natural design method. This method is not coupled with the "nite element mesh generation pre-processor. In this case, additional "ctitious static, thermoelastic or eigenvalue analyses are performed on the "nite element model and displacement boundary conditions are applied to ensure that geometric variations of the model satisfy the shape constraint design. As its principal drawback is its di$culty to be use due to the designer must still de"ne the appropriated "ctitious loads/boundary conditions and the optimal shape it is not readily interpreted by the designer, since it is de"ned by deformation "elds on mode shapes. Kodiyalan et al. [9], Chen and Tortorelli [10] used a method where a solid model of the structure to be optimized is used instead of a "nite element model or a "ctitious analysis over it. The use of a solid model is advantageous since it is an integral part of CAD and the design variables are the solid model dimensions, which are easily interpreteted by the designer. The solid modeling method also facilitates the de"nition of the "nite element model and the optimization problem through automatic mesh generation and associativity between the solid and the "nite element model. The approach proposed here is based on the parameterization techniques. In order to overcome its drawbacks a friendly interface was created. The user can easily de"ne the optimization model by using an automatic mesh generator. The dimensions of the model also can be handled as parameters. The mesh degradation is avoided by de"ning constraints over the topology of the mesh and by the use of side constraints On the other hand, the geometric modeling in CAD systems can be carried out by using Lagrange polynomials, Bezier curves, B-splines and b-splines curves and surfaces and Coons patches (see for instance [11}13]). Among these approaches, the b-spline curve properties can be

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easily controlled by using its parameters. Other advantages that encourage the use of this technique are, the curve lies in a convex hull of the vertices, the curve does not depend from an a$ne-transformation and the tangents of the initial and "nal points are de"ned by the "rst and the last edge, respectively, of the polygon. The above characteristic allows the imposition of C continuity [11]. And "nally, the search of a robust optimization algorithm, with good balance between e$ciency and e$cacy, necessary to survive in many di!erent environments, has led to the use of Genetic Algorithms (GAs). Genetic Algorithms have many advantages over traditional search methods. Among other considerations, they do not need further additional information than objective function values or "tness information. This information is used to determine the success or failure of a design in a particular environment. Moreover, its ability to "nd out the optimum or quasi-optimum solution gives it a privileged position as a powerful tool in non-conventional optimization problems. All of these advantages have led to propose the combination of GAs with geometric modeling, by using b-splines curves, in order to solve non-conventional problems like shape optimization problems. In a previous work [14], the authors have shown that the combined approach of GAs and FEM provides a powerful tool to optimize 2D bidimensional models. However, the nodal coordinates in the FEM model has proven to be somewhat di$cult to control, which suggested the use of a class of spline-curves to model the boundary variation.

2. Shape optimization methodology using genetic algorithms The optimization technique used in this work was the Genetic Algorithms due to its great versatility, easy implementation and its ability to "nd out the optimum or quasi-optimum solution. A complete description of GAs techniques can be found elsewhere [15}17] among other excellent works available in the technical literature. Genetic Algorithms are search algorithms based on the mechanics of natural selection and natural genetics. They were developed in an attempt to simulate some of the processes in natural evolution. Those algorithms converge quickly to the optimum structure with a minimum e!ort, having to test only a small fraction of the design space to "nd out either the near optimum or the optimum solution. The main principle of GAs was early stated by Holland [15]. It was based on the existence of two simultaneous processes: the natural one and the implicit one. In the natural process, a computation proportional to the size of the population &n' is performed at each generation, but in the implicit process, the algorithm carries out a more e$cient searching, which is proportional to n, needing no further information than that contained in the population itself. In order to use GAs in optimization problems, some parameters of interest in the system to be optimized have to be chosen. These parameters are called design variables. In this work the following parameters were used: E the position of the b-spline curve control vertices (< ), N E the shape parameters b and b , and   E the parametric dimensions to de"ne the optimization model (D ). K

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Then, they are represented by some set of strings coded in binary or other codes, which correspond to the chromosomes of living things (see, e.g., [18,19]). In this case, an individual design is represented by a binary string of appropriate length incorporating, generally by simple concatenation, the values of all design variables: Design"1b b < < 2< D D 2D 2   N N NL K K KL Chromosome"11 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 1 02

(1)

These strings form the initial population. Once the population has been de"ned, a "tness function or objective function that measures the behavior of each individual in its environment has to be de"ned. This function provides a direct indication of the performance of each individual to solve the optimization problem subjected to the imposed constraints from the environment. With the population ranked according to the "tness, a group of chromosomes are selected from the population. There exist several methods to select parents. The following can been used here [20] E Stochastic sampling with replacement (SSWR). E Remainder stochastic sampling without replacement (RSSWR). E Tournament selection (TS). Using the crossover and mutation operators, the selected chromosomes are then reproduced. The crossover operation consists in taking two selected chromosomes as parents. Then, they are either crossed by using a certain probability value in order to create two new chromosomes (children) or they are directly included into the new population. Many procedures can be found to carry out this task in the technical literature [17]. In the present work the following criteria were implemented: simple crossover, double crossover and uniform crossover. The mutation operator gives each bit in a chromosome the opportunity of changing (if it is 1 to 0 or vice versa). The selection of the genetic operators depend upon the problem to be optimized. In the examples presented in this work, the RSSWR selection scheme was used. A double/uniform crossover technique was used since these techniques lead to a better combination of the chromosomes than other simple operators do, and the direct mutation is also used. The values of probability of crossover and mutation were chosen from those suggested by Goldberg [16]. The optimization scheme can be summarized as follows: 1. 2. 3. 4.

Select parents. Apply genetic operators in order to create the next generation. Evaluate objective function and constraints. If the optimum has not been reached yet, repeat steps 1}4 until the optimum is found or the process will be stopped by end criteria.

The bulk of Genetic Algorithms processing power is due to the simple transformation carried out by selection and crossover operators. Mutation plays a secondary role in the operation of genetic algorithms and it is needed because even though selection and crossover e!ectively search

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and recombine extant notions, occasionally they may become overzealous and they can lose some potential useful genetic material (1s or 0s at particular locations). Thus, the mutation operator protects against such irrecoverable premature loss of important notions. In the technical literature, numerous advanced operators can be found. Moreover, each user can de"ne his own operators suited to their speci"c problems. The operators elitism, rebirth, niche and speciation can be also used with Genosoft System. A more detailed explanation about the use of these operators can be found in Ref. [14]. 3. b-splines curves formulation The b-spline curves are geometric design elements, which stem from the well-known B-spline curves. This new formulation [21}23] de"nes additional parameters that control the bias (b ) and  tension (b ) of each curve segment. The e!ect of these parameters is to change the parametric  continuity between the curve segments while maintaining the geometry continuity. The parametric equations of b-splines can be visualized as the path of a particle moving through space. Increasing b above unity, the velocity of the particle immediately after a knot point  increases. This serves to push it further in the direction of travel before it turns as in#uenced by the next control point. This is said to bias the curve to the right. Decreasing b below unity, the particle  velocity decreases and thus, it biases the path towards the left. The parameter b controls the  tension in the curve. As b is increased above zero, the knot points are pulled towards their  respective control points. For negative b , the knot points are pushed away. There are two types of  formulation: E Uniformly shaped b-splines: where the parameters b and b have the same value along the   entire curve. E Continuously shaped b-spline: where b and b can have di!erent values among the di!erent   curve segments. The b-spline curves are de"ned with a set of points called control vertices (< ). Although these K points are not part of the generated curve, they completely de"ne its shape. The vertices of a curve are sequentially ordered and connected in succession to form a control polygon (Fig. 1). In this way, the b-spline curve approximates a series of m#1 control vertices with m parametric curve segments, Q (u). The coordinates of a point Q (u) on the ith curve segment are then given by G G  Q (u)" b (b , b ; u) < 0)u(1, i"0 ,2 ,m!1. (2) G P   G>P P\ One of the most important advantages of this formulation is the local control. This control is obtained by the exploitation of the piecewise representation of the b-splines formulation, which is based in a local basis; that is, each b-spline basis function has local support (nonzero over a minimum number of spans). Since each control vertex is associated with a basis function, it only in#uences a local portion of the curve and it has no e!ect on the remaining part of the curve. Then, when moving a single control vertex, a localized e!ect on a predetermined portion of the curve or surface is obtained. In order to get local control, a b-spline curve segment is completely controlled by only four of the control vertices; therefore, a point in this curve segment can be regarded as

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Fig. 1. b-splines control polygon with end vertex interpolation.

a weighted average of these four control vertices. Associated with each control vertex is a weighting factor. The weighted factors b (b , b ; u) are the scalar-valued basis function, evaluated at some P   value of the domain parameter u, and of each shape parameter b and b . If b '0 and b *0 then     they form a basis; that is, they are linearly independent, and any possible b-spline curve segment can be expressed as a linear combination of them. Each basis function is a function of b and b and   the u such that it is a cubic polynomial in u whose polynomial coe$cients are themselves functions of b and b :    b (b , b ; u)" C (b , b )uE for 0)u(1 and r"!2, !1, 0, 1. (3) P   EP   E Now, by applying the geometric continuity constraints to the joint of the ith and (i#1)th curve segments, the following conditions yield: Q (0)"Q (1), G> G Q (0)"b Q(1), G>  G Q (0)"bQ(1)#b Q(1). (4) G>  G  G One more constraint is required in order to uniquely determine the coe$cients functions C . EP A useful constraint for axis independence and convex hull properties is to normalize the basis functions (that is, their sum is the unity), at u"0: C (b , b )#C (b , b )#C (b , b )#C (b , b )"1. (5) \    \         The "rst and second derivative vectors can be easily obtained from Eq. (2). By substituting these expressions in Eqs. (4) and (5), a system of linear equations follows. The solution of this system gives the C coe$cients. A detailed discussion of b-spline curves formulation can be found in Refs. EP [21,22]. The complete de"nition of a curve by an open b-splines formulation requires the speci"cation of an end condition. Di!erent techniques can be used to solve this problem. In this paper, the

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technique of phantom vertices was used. In this technique, an auxiliary vertex is created at each end of the control polygon. The auxiliary vertices are created for the sole purpose of to de"ne the additional curve segments, which satisfy some end conditions. As these vertices are unaccessible to the user they are not displayed; thus, they will be referred to as phantom vertices. It is frequently desirable and convenient to constrain the initial and terminal positions of the curve to coincide with the initial and terminal vertex, respectively; that is, the curve starts at < and M ends at < . K The phantom vertices can be obtained by using the following expressions: 1 < " [< !< ]#< ,   \ b   < "b[< !< ]#< . K>  K K\ K

(6)

4. The optimization process as a whole The main #ow chart of the optimization process carried out by the GENOSOFT SYSTEM [19] is displayed in Fig. 2. The "rst stage is the GENOPRE module. It is located outside the iterative loop and it is the only stage in which the user interaction is needed. The geometric model generation is now consider. In order to create this model the user has to divide the structure in super- elements and has to de"ne the number of divisions in each direction of each super element, since the mesh generation module needs this information. Also in this module, the user has to de"ne the structure boundaries using b-splines. The b-spline curves are de"ned by a set of key points. Some of the coordinates of these key points will be the design variables, which may or may not be independent. The other design variables that can be de"ned are: the shape parameters (b and b ) of the b-spline curves and the dimensions of the   structure that the user de"ned as parametric design variables (D ). K Another new module is the mesh generation. In this module an automatic mesh generator is used to create a valid and complete "nite element model (or analytical model). Another function of this module is to transmit the information of the geometric model to the analytical one. The other modules are the same as described before (see Ref. [19]) The shape optimization problem consists in to "nd out the shape of the model which has a minimum weight (it could also be used either the piece area or the piece volume) and the minimum stress concentration zones. So that, the objective function is stated by , min (w)" < o , (7) G G  where < , o are the Volume and density of element i, Nel the number of "nite elements of the G G model, subject to: (a) Stress restrictions: The Von-Mises stresses, calculated at Gauss points of the "nite element, must not exceed the limit value p :  (8) p G!p )0.  

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Fig. 2. Main #ow chart of the optimization process.

The Von-Mises stress is calculated as p "[(p !p )#3q ]. (9)  V W VW (b) Nodal coordinates restrictions over the control vertices (< ): nodal coordinates of some nodes of N the control polygon should not move beyond certain limit values in X and > directions in order to avoid geometry distortions. (c) Restrictions on the shape of the elements of the moving boundary, in order to avoid singular or negative Jacobians.

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(d) Side constraints: This kind of constraints depend on the problem to be optimized and they are used to prevent topology changes in the con"guration of the geometric model which may yield non-real geometries. In order to incorporate the constraints described above, the penalty method is used. In this method the "tness of an individual design is increased when constraints are violated. Then Eq. (7) can be written in the following way: g "=#j"*p"#k"*
5. Numerical examples In order to demonstrate the versatility and power of the proposed shape optimization technique, two numerical examples are presented and discussed. In the following numerical examples, each problem is simulated three times from the di!erent initial populations and then each result is shown as the average of the three results. On the other hand, GA methods are usually more ine$cient (in cpu time) than deterministic methods, since they evaluate many times the objective function. However, during this former step of the research, the interest is focussed in to obtain a robust and reliable optimization technique for complex problems, instead of to reduce signi"cantly the cpu time. This is a subject of further research. 5.1. Plate subjected to tension The geometry, dimensions, boundary conditions and loading of a plate subjected to traction is depicted in Fig. 3. The objective of this problem is to "nd out the optimal surface area, providing that the Von-Mises stresses must not exceed 34.50 Mpa. Also, Fig. 3 shows the boundary AB where the b-spline was de"ned. The control polygon is de"ned by "ve vertices. Three of these vertices can move in the X direction. As shown in Fig. 3 the initial area of the plate is 61935.00 mm. Table 1 contains the genetic parameters and operators used to optimize the plate. The optimized shape of the plate is shown in Fig. 4. It can be noted how the plate area was reduced according to the stress constraints. The "nal area is 45632.00 mm, which means a 26.32% reduction. The same problem was previously solved by the authors, by using some mesh nodal coordinates of the analytical model as design variables, as described in Ref. [14]. In that case, the number of moving joints was 64 and it required eight design groups. On the contrary, in the case shown in

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Fig. 3. Plate with b-splines boundary (A}B), geometry, dimensions, loads and boundary conditions.

Table 1 Genetic parameters for plate under tension Population size Number of generations Selection scheme Crossover scheme Crossover probability Mutation probability

100 50 RSSWR Double crossover 0.8 0.005

Fig. 4 the number of moving joints was three and three design groups were used. This group reduction was mainly due to the use of a geometric model, and the way of de"ning the moving boundary (A}B). In both cases similar results were obtained, as shown in Table 2. The initial and "nal stresses are shown in Figs. 5 and 6, respectively. Note how the shape is smoothed by the splines curves and its "nal con"guration is the result of the interaction between allowed stress and the element stresses in the moving zone of the piece. Fig. 7 displays the evolution curves of the stresses and weight. It can be noted that the non-dimensional weight factor is always below one, which clearly show the stability of the proposed optimization method. The total process stopped after 5000 FE evaluations, which took about 150 min, including IO operations, in a 233 MHz Pentium-Pc based computer.

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Fig. 4. Final shape of the optimized piece.

Table 2 Comparison between using joint nodes or b-splines as optimization technique Optimization technique

Final area (mm)

Final maximum stress (Mpa)

Joint nodes [14] b-splines (present work)

45665.00 45632.00

34.13 34.20

5.2. Connecting rod The goal of this problem is to show the ability of the algorithm to handle real-life problems. For this purpose the well-known application of a connecting rod was chosen, which is designed to minimize the surface area and to maintain the structural integrity. The "nite element model of the rod is shown in Fig. 8. This model was de"ned with 232 "nite elements and 225 nodes, having an initial area of 3622.37 mm. The di!erent design variables used to optimize the piece are de"ned as: h , h , h , h , r , r , r and the external boundary AB was modeled using b-splines.        The boundary conditions are a distributed force at the pin-end and "xed conditions at the crankshaft end. The total distributed forces applied on the pin-end are 12.000 N in the x-direction and 10.000 N in the y-direction. So, the half-symmetry model was used (6.000 N, 5.000 N). The boundary conditions, including those necessary for symmetry, and the "nal loading are presented in Fig. 9.

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Fig. 5. Initial stresses for the plate subjected to traction.

Fig. 6. Final Von-Mises stresses for the plate.

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Fig. 7. Genetic evolution of stresses and weight (plate subjected to traction and design variables de"ned with b-splines).

Fig. 8. Connecting rod geometry and dimensions(mm). Design variables and moving external boundary A}B.

Fig. 9. Boundary conditions and loading for connecting rod.

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Table 3 Side constraints used in the optimization of a connecting rod Constraint number

Side constraints

1 2 3 4 5

1!h /20)0  1!3h /(r #2r )    1!3h /(r #2r )    1!r /12)0  1!h /(r #1))0  

Table 4 Genetic operators and parameters used in the optimization of a connecting rod Population size Number of generations Selection scheme Crossover scheme Crossover probability Mutation probability

150 55 RSSWR Uniform crossover 0.8 0.005

Fig. 10. Final shape of the optimized connecting rod.

The material properties were Young's modulus "2.07;10 Mpa and Poisson's ratio"0.30. The cost function is speci"ed as the area and the constraints are prescribed to limit the Von-Mises stresses throughout the connecting rod, which must be less than 720 Mpa. The side constraints used in order to maintain structural integrity are collected in Table 3. These constraints prevent against designs that will produce no realistic geometric models. The genetic operators and parameters used to optimize the structure are displayed in Table 4. The optimal design, shown in Fig. 10, yields a 51.75% reduction of the surface area. Both initial and "nal stresses are displayed in Figs. 11 and 12. It can be observed how the maximum Von-Mises stresses move from 421.15 MPa in the initial shape to 710 MPa in the optimal shape, which is below the maximum allowed value 720 MPa.

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Fig. 11. Initial Von-Mises stresses distribution (connecting rod).

The optimization history, illustrated in Fig. 13 for the non-dimensional weight factor (best area /initial area) and the non-dimensional stress factor (maximum stress /allowable stress), shows G G convergence in 35 generations, it is to say at 5250 "nite element analysis. Moreover, the weight factor is less than one and it behaves in very stable manner around the optimum value. This is a clear indication that the algorithm is stable and it is able to deal with complex shape optimization problems. In this example, the total process stopped after 8250 FE evaluations, which took about 250 min, including IO operations, in a 233 MHz Pentium-Pc based computer.

6. Concluding remarks In this work, the combination of two powerful tools in optimization "eld, geometric modeling and genetic algorithms, is presented. As shown in the numerical examples, the use of this combined approach improves the "nal results and saves computer time, since few design variables are required. Also, in order to get these improvements, an automatic mesh generator was used to generate the analytical model from the geometric model through a set of geometric parameters. The use of genetic algorithms as optimization technique improves the performance of the approach, due to its great advantages as compared with traditional optimization techniques. One of the most attractive points of this technique is that it requires no calculation of sensitivities and it reaches an optimum or quasi-optimum solution. Further e!orts are being done to apply this approach for the optimization of 3D FEM complex models. Another future application of this approach would be the use of boundary elements instead

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Fig. 12. Final stresses distribution (connecting rod).

Fig. 13. Genetic evolution of stresses and weight (connecting rod).

of "nite elements, in order to overcome the mesh degradation discussed before and to reduce the number of active constraints.

Acknowledgements The authors wish to acknowledge the support provided by the Council for Humanistic and Scienti"c Development (CDCH) and National Council for Scienti"c Research (CONICIT) of Venezuela

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