A NEW APPROACH TO MINIMIZE THE DISTORTION OF QUADRILATERAL AND HEXAHEDRAL MESHES

Transcription

1 European Congress on Computational Methods in Applied Sciences and Engineering ECCOMAS P. Neittaanmäki, T. Rossi, K. Majava, and O. Pironneau (eds.) O. Nevanlinna and R. Rannacher (assoc. eds.) Jyväskylä, 8 July A NEW APPROACH TO MINIMIZE THE DISTORTION OF QUADRILATERAL AND HEXAHEDRAL MESHES Josep Sarrate and Antonio Huerta Laboratori de Càlcul Numèric, Departament de Matemàtica Aplicada III, ETSE de Camins, Canals i Ports de Barcelona, Universitat Politècnica de Catalunya, Edifici C, Jordi Girona -, E-8 Barcelona, Spain jose.sarrate@upc.es, web: Key words: Finite element method, mesh generation, mesh smoothing techniques, unstructured meshes, quadrilateral elements, hexahedral elements. Abstract. In this paper a continuous minimization of the mesh distortion metric proposed by Oddy for quadrilateral and hexahedral elements is presented. Although it has been extensively used, the original definition has some limitations that preclude its use in a continuous minimization procedure. For instance, it is only valid for convex quadrilaterals, or it gives an infinite distortion value for a degenerated quadrilateral with triangular shape (similar behavior is observed for hexahedral elements). In order to overcome these drawbacks, we deduce a geometrical interpretation of the original distortion metric, both for quadrilateral and hexahedral elements. Based on this interpretation, first we develop a new alternative to compute the distortion metric for non-convex quadrilateral and hexahedral elements. Then, a continuous minimization algorithm of the improved distortion metric is presented. It is important to note that the original and the improved definition of the distortion metric coincide around the optimal solution. Finally, some numerical examples are presented to asses the efficiency of this algorithm.

2 INTRODUCTION The finite element method has became one of the most powerful and versatile computational technique to solve partial differential equations. However, it is hampered by the need to generate an adequate discretization. It is well known that the geometry of the mesh is one of the most important factors influencing the accuracy of the finite element solution []. Quadrilateral and hexahedral mesh generation algorithms [,,, 5, 6, 7] may initially yield meshes with very distorted elements. Therefore, mesh quality enhancement procedures are needed in order to improve the overall mesh quality. There are two basic ways to reach this goal. The first one, often called make-up techniques, is focused in the improvement of the mesh topology. The second one, called mesh smoothing, improves the shape of the elements by modifying the position of the inner nodes once the topology is fixed. Concerning the distortion of the mesh elements, the former plays a minor role because. Therefore, special attention has to be focused on the smoothing algorithm. Several smoothing algorithms has been developed in the last decades. For instance, [8] extend the scope of the variational methods, widely used for structured grids, to non structured triangular meshes. Other commonly used smoothing technique for unstructured meshes is the so-called Laplacian method [9, ]. Giuliani [] developed a new rezoning algorithm based on a geometrical criteria. In this method the position of every node is modified in order to minimize the average of the geometric-oriented distortion of elements meeting on it. Some modifications can be added to the original algorithm in order to maintain the prescribed element size []. However, any of the last three methods minimize a distortion measure of the quadrilateral or hexahedral elements. Oddy et. al. [] developed a distortion metric for quadrilateral and hexahedral elements. It measures the degree of deviation of an element in the physical domain respect to its shape in the computational domain. However, it is only valid for convex elements. This limitation precludes the use of this distortion metric in an continuous minimization algorithm. For instance, in a mesh generation algorithm, non convex quadrilaterals may appear before any mesh quality enhancement procedure is applied in order to improve the overall mesh quality. To overcome this limitation, first we introduce a geometrical interpretation of the original distortion metric. Based on this interpretation, second we extend the distortion metric to non convex quadrilateral and hexahedral elements. Third, a continuous minimization procedure of the improved distortion metric based on a Newton Raphson algorithm is developed. Finally, numerical examples show that the original and the improved definition of the distortion metric coincide around the optimal solution. DEFINITION OF THE DISTORTION METRIC Oddy et. al. [] proposed a point wise distortion metric for quadrilateral and hexahedral elements. This metric is based on the Jacobian of the bilinear (trilinear for D

3 analysis) iso-parametric mapping. The basic idea of this metric is to measure the degree of deviation of the element in the physical domain respect to its shape in the computational domain. It accounts for both shearing and stretching effects. Moreover, it is not affected by rigid body motions (displacements and rotations), and is independent of the element size. It is defined as: D Oddy = C : C (, tr (C)) () n where n = or n = for D or D problems respectively, and C = J T J, () J = J, J () being J the Jacobian matrix of the isoparametric mapping. Notice that four node linear isoparametric elements are assumed (eight node elements for D analysis). As noted by Oddy et. al., () is a sensitive measure of the distortion of the element, since it is a function of the Jacobian to the fourth power. NUMERICAL EVALUATION OF THE DISTORTION METRIC FOR QUADRILATERAL ELEMENTS Equation () is a point wise measure of the distortion in the element. However, for mesh generation and smoothing purposes, it is more convenient to obtain a global measure of the element distortion. Hence, it is necessary to assign a distortion value to each element. In this section, we first introduce a new geometrical interpretation of the distortion metric () based on its particularization to parallelograms. Second, we generalize this interpretation in order to evaluate the distortion metric () at a given point of a general quadrilateral.. Particularization to parallelograms Jacobian matrix is constant everywhere in a parallelogram. Using equation (), it is straightforward to obtain that its distortion is D Oddy = l l sin (α) + l l sin (α) + sin, () (α) where α, l and l are defined in figure. Equation () can be written as D Oddy = (Q Oddy ), (5) where Q Oddy = l + l A, (6)

4 l l Figure : Basic variables used to define a parallelogram. is called geometric efficiency, and A = l l sin(α), is the area of the parallelogram. Note that D Oddy and Q Oddy have the same sensibility to capture up any cause of distortion. However, they show a different grow slope.. Evaluation of the distortion metric at a given point From the computational point of view, equation (5) is a simple and efficient alternative to compute the distortion of a parallelogram. Therefore, our goal is to use it when computing the distortion of a general quadrilateral. First, we consider the following general property of the bilinear iso-parametric mapping. Consider a square element in the computational domain. The image of a straight segment parallel to one of its sides is a straight segment in the physical domain (see figure (a)). Following this property, the concept of the local parallelogram at a given point P of a general quadrilateral is introduced. To this end, let p be a point in the computational domain and let P be its image in the physical domain. Let aa and bb be two segments that cross at point p and are parallel to the sides of the reference square in the computational domain (see figure (a)). Let AA and BB be their image by the iso parametric mapping. Then, the local parallelogram at point P is defined as the parallelogram limited by sides PA = AA and PB = BB (see figure (b)). That is, the sides of the local parallelogram at a given point P are the image of the segments, parallel to the sides of the square of reference, and that cross at point p. The final result of this subsection is that, the distortion () at a given point P of a quadrilateral can be computed as the distortion of the local parallelogram defined at that point (a detailed proof is presented in [5]). Moreover, this result has been extended to hexahedral elements in [6]. This geometrical interpretation of the distortion () allows to derive a new alternative to compute the distortion of an element.. Numerical evaluation of the element distortion In order to define how to compute numerically the distortion of an element, two subjects have to be addressed. First, it must be specified at which points of the element the

5 b B p a a A P A b (a) B B B A P A A B (b) Figure : (a) The segments aa and bb are parallel to the sides of the square element in the computational domain. Their transformation by the bilinear iso-parametric mapping are two straight segments: AA and BB. (b) Definition of a local parallelogram at a given point P. Note that PA = AA and PB = BB. distortion has to be evaluated; and second, how these values contribute to the global element distortion. The most common choice is to evaluate the distortion of the element at its vertices [, ]. If the geometrical interpretation of the distortion D Oddy introduced in equation (5) is used, then the distortion at the vertices of the element can be calculated via the local parallelogram defined at each vertex. Therefore, at any vertex i, for i =,...,, the distortion of the local parallelogram is: l i = (x [i+] x i ) + (y [i+] y i ), l [i+] = (x [i+] x i ) + (y [i+] y i ), A i = (x [i+] x i )(y [i+] y i ) (x [i+] x i )(y [i+] y i ), (7) Q Oddy,i = l i + l [i+] A i, D Oddy,i = (Q Oddy,i ), where (x i, y i ) are the coordinates of vertex i, and [i] = mod(i, ) + (the remainder plus one of i when it is divided by ). Note that, from equation (7), the distortion 5

6 B B (a) A P A B (b) A Figure : (a) Quadrilateral () and the quadrilateral defined by its Gauss points ( ). (b) Trapezoidal element () with a null distortion at its center, P. The local parallelogram associated to its center is (PA B ). at a given vertex does not depend on the position of the opposed vertex. For instance, the distortion on node of figure (a), does not depend on the position of node. In addition, the computational cost of equation (7) may be significantly reduced if the length of the adjacent sides is already stored (which is a common strategy in quadrilateral mesh generation). The distortion of an element may be also evaluated at the Gauss points or at the center of the element. If the Gauss points are used, it is straightforward to show that the distortion of an element evaluated at its Gauss points is equal to the distortion of the element defined by the Gauss points of the original element, and evaluated at its vertices, see figure (a). Therefore, the same procedure (7) can be used to evaluate the distortion of an element at the Gauss points. In general, the values of the distortion at the Gauss points are smaller than the values at the vertices. Hence, a smoother variation of the distortion is obtained. The last choice is to evaluate the distortion of an element at its center. This is the simplest choice from the computational point of view. However, it is not a correct measure of the geometric quality of a quadrilateral. The same drawbacks that appear when the Gauss points are used to evaluate the distortion of an element are also present in this case (note that the center of an element can be understood as a Gauss point). Moreover, there are very distorted elements with null distortion at its center. For instance, figure (b) shows a trapezium and the local parallelogram associated to its center. As it can be appreciated, the local parallelogram is a square. Therefore, D Oddy = if it is evaluated at the center of the trapezium. In order to define how the point values contribute to the global element distortion, it is usual to define the distortion of an element as the maximal nodal distortion [,, ] Oddy = max i=,..., (D Oddy,i). (8) Note that if the maximum value of the distortion is achieved at a given vertex, according to equation (7), the distortion of the element will be independent of the position of the 6

7 opposed vertex. Therefore, the opposed vertex can be moved without modifying the element distortion (until the vertex of maximum distortion is changed). Moreover, a continuous minimization of distortion (8) is not possible because it is not continuous and differentiable. In order to overcome these drawbacks, the distortion of an element is defined as the mean value of the distortion at the vertices of the element [] Oddy = i= Oddy,i, (9) where is the number of element nodes, and Oddy,i is the value of the Oddy distortion (7) at the i-th node of the element e. In general, equation (9) will generate smaller values than equation (8) since it is a mean value. However, it is always differentiable and all vertices contribute to the distortion of the element.. Numerical evaluation of the mesh distortion It is a common choice [9,,, ] to define the mesh distortion as the sum of the distortion of its elements D mesh = N elem e= Oddy, () where N elem is the number of the mesh elements. If the distortion of the element is evaluated according to (9), we obtain D mesh = N elem e= Oddy,i = i= N lp k= D Oddy,k = N lp (Q Oddy,k ), () where is the number of element nodes ( for a bilinear quadrilateral or 8 for a trilinear hexahedral element), and N lp is the total number of local parallelograms in the mesh (N lp = N elem ). The factor / in equation () can be understood as a scale factor..5 Limitations of the Oddy s distortion The distortion metric () has been extensively used in mesh generation algorithms [,, 5, ]. However, if it is used in a continuous minimization algorithm of the distortion of a mesh [], two severe limitations appear. The main reason of these limitations is that distortion metric () is only valid for convex and non-degenerated quadrilaterals. Note that this limitation does not depend on the point where the distortion metric is evaluated. Therefore, if distortion metric () is used to compute the mesh distortion () of a given quadrilateral grid, both infinite asymptotes and local minimums may appear. In order to visualize this behavior, consider the mesh presented in figure (a). It is composed by two unit squares and one degenerated element with triangular shape. The target is 7 k=

8 (a) (b) 8 6 D Y (c) (d) Figure : (a) Initial mesh with degenerated elements. (b) Smoothed mesh with minimum Oddy distortion. (c) Mesh distortion values when node 5 moves from to 7. (d) Mesh distortion contour levels when boundary nodes are fixed. to minimize the mesh distortion, D mesh, taking into account that the boundary nodes are fixed and it is only possible to move the inner node (node 5). Figure (b) shows the optimal solution. Figure (c) shows the values of the mesh distortion, D mesh, when the inner node 5 moves along the symmetry axis (from node to node 7). Note that a vertical asymptote appears when a degenerated element with triangular shape is achieved (this situation corresponds to figure (a)). Therefore, a configuration with a non-convex quadrilateral (Y 5 > ) may exists with the same mesh distortion than other composed exclusively by convex elements. Moreover, a local minimum appears at Y 5 =. which does not have a real meaning. Figure (d) shows the contours plots of the mesh distortion, D mesh, when the inner node moves inside the domain. Note that, apart from the global and local minimums that are detected in figure (c), other local minimums appear. NUMERICAL EVALUATION OF THE DISTORTION METRIC FOR HEXAHEDRAL ELEMENTS Similar to section, the distortion metric () can be also computed from the geometric properties of a given hexahedral element in the physical space. To this end, the first step is to evaluate () for a parallelepiped. The second step is to compute the distortion metric () in a given point of a hexahedral element in terms of the local parallelepiped defined at this point. 8

9 . Particularization to parallelepipeds Since the Jacobian matrix is constant everywhere in a parallelepiped, it is straightforward to prove that (( D Oddy = l + l + l ) ) (A V + A + A ), () where l, l, l, A, A, A are defined in figure 5(a), and V is the volume of the parallelepiped. Moreover, the geometric efficiency for these elements is Q Oddy = (( l + l + l) (A + A + A ) + V V ). () Hence, the relationship between the distortion metric and the geometric efficiency for parallelepipeds is. Local parallelepiped at a given point D Oddy = (Q Oddy ). () The definition of local parallelepiped at a given point of a hexahedral element is analogous to the definition of local parallelogram at a given point of a quadrilateral element. Then, it is proved (see [6] for details) that the distortion metric () at any point of a hexahedral element can be computed in terms of the local parallelepiped defined at that point. Moreover, the distortion of a hexahedral element can be computed in terms of the distortion at its vertices according to (9), being the distortion of the i-th vertex, for A i A ii l A l i l A ii i l i l (a) A i A ii l i (b) i Figure 5: (a) Variables that determine a parallelepiped. (b) Right-hand criterium to numerate nodes, edges and sides that meet on node i. 9

10 i=,...,8 l i = (x i x i ) + (y i y i ) + (z i z i ), l i = (x i x i ) + (y i y i ) + (z i z i ), l i = (x i x i ) + (y i y i ) + (z i z i ), A ii = ((y i y i )(z i z i ) (z i z i )(y i y i )) + ((x i x i )(y i y i ) (y i y i )(x i x i )) + ((x i x i )(z i z i ) (z i z i )(x i y i )), A ii = ((y i y i )(z i z i ) (z i z i )(y i y i )) + ((x i x i )(y i y i ) (y i y i )(x i x i )) + ((x i x i )(z i z i ) (z i z i )(x i y i )), (5) A ii = ((y i y i )(z i z i ) (z i z i )(y i y i )) + ((x i x i )(y i y i ) (y i y i )(x i x i )) + ((x i x i )(z i z i ) (z i z i )(x i y i )), V i = ((y i y i )(z i z i ) (z i z i )(y i y i )) (x i x i ) + Q Oddy,i = ((x i x i )(z i z i ) (z i z i )(x i x i )) (y i y i ) + ((x i x i )(y i y i ) (y i y i )(x i x i )) (z i z i ), (( l i + li + li) D Oddy,i = (Q Oddy,i ), (A ii + A ii + A ii) + V i ) V i, where l i, l i, l i, A ii, A ii y A ii are defined in figure 5(b), and V i is the volume of the local parallelepiped associated to node i. Finally, the mesh distortion for hexahedral elements is defined according to (). Furthermore, taking into account () the distortion of a mesh composed by hexahedral elements can be computed by D mesh = N elem e= Oddy,i = i= N lp k= D Oddy,k = N lp (Q Oddy,k ). (6) Note that if distortion metric () is used to compute the distortion of all-hexahedral meshes, both infinite asymptotes and local minimums may also appear. 5 MODIFICATION OF THE DISTORTION METRIC In this section a modification of the distortion metric () is developed in order to overcome the drawbacks analyzed in the two previous sections. The main goal of the developed modification is to maintain the original definition while it is possible, and modify k=

11 Figure 6: Geometric definition for the quadrilateral test. it where the original definition can not be computed properly. This modification will be presented for quadrilateral meshes, but it is straightforward to extend it to hexahedral elements (see [6] for details). To this end, a new measure of the distortion of the local parallelogram associated to a node i, i =,...,, of a given quadrilateral element is introduced: q Oddy,i = Q Oddy,i = A i li +, (7) l [i+] where all variables have been introduced in equation (7). Note that, according to (5), the distortion metric () evaluated at node i and q Oddy,i are related by ( ) D Oddy,i =. (8) q Oddy,i Therefore, the global function () to minimize over the mesh can be written as D mesh = N lp D Oddy,k = N lp ( ). (9) k= k= q Oddy,i In order to develop the modification of the distortion metric (), a test based on the quadrilateral element of figure 6 is considered. In this test nodes, and are fixed, and node is allowed to move along x-axis. Figures 7(a) and 7(b) shows the values of D Oddy,i and q Oddy,i respectively. Since q Oddy, is a continuous function and assigns different values to different quadrilateral elements (D Oddy,i depends upon the square of q Oddy,i ) the modification of the distortion metric is based on q Oddy,i. The first step is to change equation (8) in order to eliminate an infinite value of D Oddy, when q Oddy, becomes null. To this end, D Oddy,i is approximated by a tangent parabola for values of q Oddy,i < q according to D ( Oddy,i ) if q Oddy,i q, D OM,i = q qoddy,i 8 q q Oddy,i + 6 () q if q Oddy,i < q,

12 D / q.7 x.7 x (a) (b) D_Oddy D 5 5 q_oddy (c) x (d) Figure 7: (a) Values of D Oddy,. (b) Values of q Oddy,. (c) D OM, versus q Oddy, (solid line) and D Oddy, versus q Oddy, (doted line). (d) D mesh computed using D Oddy,i (solid line) and D OM,i (doted line). where q is a fixed value. Figure 7(c) shows a plot of D Oddy, and D OM, versus q Oddy,. Figure 7(d) presents a plot of the mesh distortion using D Oddy,i and D OM,i for the test presented in figure 6 as node moves along x-axis. Note that no vertical asymptote appears if () is used. The second step is to add a term to the definition of the element distortion such that the new definition:.- depends on the same variables of q Oddy,i, for i=,...,;.- is continuous and derivable; and.- has a minimum value at the same point, or near the same point, that D Oddy,i. The chosen function is E = Ei, i= where E i = (x [i+] x [i] )(x [i+] x [i] ) + (y [i+] y [i] )(y [i+] y [i] ), for i =,...,, is the scalar product of the vectors defined by the edges that meet node i. Finally, the mesh distortion is defined taking into account the number of elements with an inner angle greater than π, denoted by n conc, according to

13 Josep Sarrate and Antonio Huerta 8 D 6 D x...6 y5 y (a) x5 (d) 6 D 5 D y y5 x x5 (b) (e) 5 D 8 6 D 5 y y5 x x5 (c) (f) Figure 8: Surface representation of: (a) DOddy ; (b) DOM ; and(c) DOM + E for the quadrilateral test of figure 6. Surface representation of: (d) DOddy ; (e) DOM ; and(f ) DOM + E for the mesh presented in figure (a).

14 where Oddy D mesh = N elem e= N elem e= is computed according to (9), and Oddy if n conc =, OM if n conc >, () OM = Nen Nen OM,i + i= i= E (e) i. Figure 8 shows the surface plots of D Oddy, D OM and D OM + E for the quadrilateral test of figure 6, and for the mesh presented in figure (a). Note that if D OM is used no vertical asymptotes. Moreover, if the mesh distortion is computed using D OM + E local minima also disappears and the global minimum is more defined. 6 MINIMIZATION ALGORITHM In this section a continuous minimization algorithm of the distortion of the mesh is presented. Since the gradient vanishes at the minimum value, the Newton Raphson method is used to cancel the gradient of the mesh distortion. Furthermore, some modifications are added in order to increase its robustness and rate of convergence. If the initial approximation (for instance, the given grid in a mesh generation algorithm) is distant of the optimal solution (the mesh that minimize the distortion metric D mesh ), the Newton Raphson method may not converge. Hence, a minimization method based on the steepest descent method with a line search strategy is developed. Our goal is to minimize the mesh distortion (). Hence, we impose: D mesh = = D mesh a i = for i =,..., N eq () where a i are free x and/or y coordinates of the mesh nodes; and N eq is the total number of free coordinates, which corresponds to the total number of equations in the non linear system (). The non linear system () is solved in an iterative procedure using the Newton Raphson method: H k D mesh a k+ = D k mesh () where H k D mesh is the Hessian matrix of D mesh at iteration k; a k+ = a k+ a k, being a k and a k+ the arrays of free coordinates in two consecutive iterations; and Dmesh k is the gradient of the mesh distortion at iteration k. Note that the Hessian matrix is a N eq N eq symmetric square matrix. Moreover, if the minimization of the mesh distortion () has a

15 solution, then H Dmesh will be positive defined in a neighbor of the solution. The element contribution to the Hessian matrix and right hand vector of the system () are detailed in [6]. In order to increase the robustness of the Newton Raphson method, three modifications are added to the original algorithm:.- imposing an upper limit to the increment, a k+, will avoid large nodal displacement where small elements appear;.- implementing a line search strategy will increase the rate of convergence of the original algorithm; and.- applying the steepest descent method with a line search strategy during the initial iterations will ensure that Newton Raphson method converges to the optimal solution, a opt, even if a distant initial guess, a, is used (see details in [6]). 7 EXAMPLES In order to asses the capabilities of the minimization algorithm, four examples are presented in this section. The first two examples correspond to quadrilateral meshes and are generated using an unstructured quadrilateral mesh generation algorithm previously developed [5]. The last two applies to hexahedral meshes wich are generated using a comercial sweep algorithm. Once the meshes are generated, all of them are smoothed using the Giuliani method [], a modification of the Giuliani method [] and the method presented in this paper. For all of them, the mean and the maximum values of the distortion of the elements are presented in table. Characteristic values of the Oddy distortion metric Algorithm Example I Example II Example III Example IV D D max D D max D D max D D max Giuliani Giuliani mod Minimization Table : Mean and maximum values of the mesh distortion. The first example corresponds to the discretization of a ring-shaped domain with a hight element density prescribed on the outer boundary (see figure 9(a)). The mesh is composed by 56 nodes and 5 elements. The distribution of the element distortion is presented in figure 9(c). In order to clarify the visualization of the results, the range of distortion is limited to the gap [, ]. Hence, elements for the Giuliani method and for the modification of the Giuliani method are out of scale. Note that less distorted elements are generated with the new smoothing algorithm. Moreover, D max is reduced by one order of magnitude if the minimization algorithm is used. The second example, presented in figure 9(b), shows the discretization of a square of length with a small element size prescribed along one diagonal. The mesh is composed by nodes and 5 elements. As in the first case, the minimization algorithm 5

16 (a) (b) Oddy_min Giulliani_ori Giulliani_min Oddy_min Giulliani_ori Giulliani_min log (# elements)+ log (# elements)+,5,5,5,5 Distortion (c),5,5,5,5 Distortion (d) Figure 9: (a), (b) Smoothed meshes corresponding to examples I and II, respectively. (c), (d) Element distortion distribution for examples I and II, respectively. reduces the mean and maximum element distortion values (see table ). Furthermore, figure 9(d) shows that the distribution of the distortion of the elements is concentrated in a narrower gap of small distortion values. Figure (a) presents the smoothed mesh of a cylinder. It is composed by 59 nodes and 5 elements. As it can be observed, high distorted quadrilaterals appear in the cap surfaces. However, the developed algorithm allows to reduce in one order of magnitude the mean and the maximum values of the distortion of the hexahedral elements (see table ). Figure (c) shows the distribution of the distortion of the elements. As in the first example, the range of the distortion values is reduced to [, 6]. In this example 6 elements are out of scale when the Giuliani and the modification of the Giuliani methods are used. Figure (b) shows the smoothed mesh corresponding to the fourth example. It is composed by 6 nodes and 6 elements. As in the previous examples, better results are obtained with the new algorithm (see figure (d) and table ). 6

17 (a) (b) Oddy_min Giulliani_ori Giulliani_min Oddy_min Giulliani_ori Giulliani_min log (# elements)+ log (# elements) Distortion (c) 5 5 Distortion (d) Figure : (a), (b) Smoothed meshes corresponding to examples III and IV, respectively. Element distortion distribution for examples III and IV, respectively. (c), (d) 8 CONCLUSIONS In this paper a continuous minimization of the mesh distortion metric for quadrilateral and hexahedral elements presented in [] has been developed. To this end, a geometrical interpretation of the original distortion metric is presented. Based on this interpretation, a new alternative to compute the distortion metric at a given point of quadrilateral and hexahedral elements is derived. It is proved that this new alternative may be more efficient from the computational point of view, if some element information is already stored. The geometric interpretation of the distortion metric allows to define a measure of the mesh distortion which can be used in a continuous minimization process. Then, a continuous minimization algorithm of the mesh distortion based on a Newton Raphson method is developed. Finally, the numerical examples have shown the efficiency of the developed algorithm. In general, it obtains meshes with a mean and maximum values of the distortion smaller than those obtained using other methods. This work was partially sponsored by the Ministerio de Ciencia y Tecnología under grants DPI- and REN-95-C-/CLI. 7

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