Preconditioning the quad dominant mesh generator for ship structural analysis

: This paper presents an algorithm for the fully automatic mesh generation for the 1 ﬁnite element analysis of ships and offshore structures. The quality requirements on the mesh 2 generator are imposed by the acceptance criteria of the classiﬁcation societies as well as the 3 need to avoid shear locking when using low degree shell elements. The meshing algorithm 4 will be generating quadrilateral dominated meshes (consisting of quads and triangles) and the 5 mesh quality requirements mandate that quadrilaterals with internal angles close to 90 ◦ are to 6 be preferred. The geometry is described by a dictionary containing points, rods, surfaces and 7 openings. The ﬁrst part of the proposed method consist of an algorithm to automatically clean 8 the geometry. The corrected geometry is then meshed by the frontal Delaunay mesh generator as 9 implemented in the gmsh package. We present a heuristic method to precondition the cross ﬁeld 10 of the frontal quadrilateral mesher. Also the inﬂuence of the order in which the plates are meshed 11 will be explored as a preconditioning step. 12


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This paper is concerned with automatic mesh generation in the process of the 16 structural analysis of ships and off-shore structures. The ship structural analysis by 17 the finite element method is governed by the acceptance criteria of the classification 18 societies [1]. One of the possibilities to describe an input geometry, for a large class 19 of such structures, is by the use of a dictionary of elements. The dictionaries which 20 we will be considering in this paper consist of points, rods, surfaces, and openings. 21 The points in the dictionary describe positions where the loads (for the finite element 22 analysis) are going to be applied or the measurements are going to be taken. Rods 23 describe the panel stiffeners or pillars. The set of surfaces will consist of three types of 24 entities: web surfaces (parallelograms with one dimension much smaller than the other), 25 regular surfaces (convex quadrilaterals defined by four co-planar nodes) and warped 26 surfaces (closed loop surface defined by four corners which need not be co-planar) [2,3]. 27 The preconditioning method which we are going to describe extends and improves the 28 meshing algorithm from [4]. The algorithm from [4], called pyREMAKEmsh, is designed 29 complicates the meshing task. The requirements on a mesher can be summed up as: The elements with angles less than 45 • and more than 135 • should be avoided 49 • In the high stress areas such as e.g. web surfaces, the use of triangular elements 50 should be avoided.

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• The use of triangular shell elements should be kept to the minimum 52 • Quadrilateral shell elements with high aspect ratio as well as distorted elements 53 should be avoided -aspect ratio for quadrilaterals is to be kept close to 1 but should 54 not exceed 3 for 4 node elements and should not exceed 5 for 3 node elements 55 (triangles).

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• Web surfaces should be modeled with at least four elements allong the shorter 57 dimension, and ideally with precisely four elements.

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These requirements pose a challenge for implementing fully automatic meshing routines.

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Namely, these rules most often pose local linear constraints on the mesh and as such 60 clash with the geometry optimization algorithms which search for globally optimal 61 tessellations, see [5]. We relax those restrictions in order to accommodate more complex 62 geometries typically encountered in ship structural analysis. stiffeners are used to split a plate with which they intersect, but are not included in the 73 dictionary as 1D elements.

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In this approach it is necessary that the original triangular mesh be generated so has internal angles equal to 60 • , see [12]. With this change of norm the triangles which 86 are generated by the Delaunay algorithm will tend to be closer to right angle triangles 87 and so will be amenable to good recombination into a regular quadrilateral mesh, see   Meshing using the preconditioned gmsh algorithm Packing for Parallelograms from [13] In this paper we will present a geometry improvement algorithm which can tackle 90 geometries containing warped plates. The algorithm is based on the processing pipeline 91 which utilizes boolean operations in the 3D geometry kernel OpenCASCADE to correct 92 the geometry errors and to enforce local linear constraints. Further, we will introduce 93 a preprocessing step to make the results of the meshing algorithm repeatable and to 94 control the mesh quality.

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The makeup of the paper is as follows. We will first present materials and methods 96 where we will describe the challenges and the necessary modifications to the algorithm 97 necessary to be able to tackle warped plates. We will then present results of the applica-98 tions of these algorithm to two characteristic examples from engineering practice. In the 99 discussion section we will present statistical evaluation of the preprocessing algorithm. would lead to low quality quads (either by the aspect ratio or angle criterion). In such 119 cases, to generate full-quad meshes the full-quad recombination algorithm is used. The 120 algorithm performs a subdivision of the elements, followed by the further recombination 121 and smoothing. The smoothing is performed using the Lloyd's algorithm, and then the 122 whole sequence is repeated until all of the triangles have been matched [13]. We will 123 employ the full recombination algorithm only in the case when the returned mixed mesh 124 has more than 5% triangles. Even then, we will return both meshes and raise the flag 125 that the processing pipeline did not produce a mesh which meets all of the constraints. 126 We will see that the fully recombined mesh forces the satisfaction of requirement for 127 almost no triangles at the expense of reducing the quality of the quadrilateral mesh.

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To this end, let us rigorously define the mesh quality measures which we will be using to check the compliance with the rules of the classification societies. For a quadrilateral q with internal angles α i (q), i = 1, · · · , 4 we define the quality measure This quality measure is equal to one for the perfect rectangle and it is zero in the presence of angles which are less than zero or greater than the straight angle (a non-convex quadrangle). Let now T be the mixed tesselation consisting of quadrilaterals and triangles. The set T is a disjoint union of the set of all quadrilaterals T q and the set of all triangles T t in the mesh. To define the stopping criterion we we introduce the following measures. Here | · | marks the cardinality of a finite set. The average η for the tesselation T is defined by The percentage of quadrilaterals with angles between 80 • and 100 • in T is marked with 146 Algorithm 1 Preconditioning the Packing for Parallelograms algorithm from [13] Require: A dictionary describing the geometry and the tolerance tol. 1: Split each web surface without openings along the smaller dimension into 4 surfaces (µ = (1, 1, 1, 1)). 2: Split each web surface with at least one opening along the smaller dimension in the four strips with the ratio n 1 : n 2 : n 3 : n 4 so that all openings are contained in the middle two strips. 3: Define the openings as surfaces and introduce them into geometry using boolean operations from the CAD kernel. 4: Subdivide plates which are co-planar with rods. 5: Index surfaces so that a quadrilateral q with larger η(q) comes before the one with the smaller. 6: Index warped surfaces so that they are appended at the and of the list of surfaces. 7: Introduce Virtual Stiffeners around openings on web surfaces 8: Generate the mesh T of the geometry G using Packing for Parallelograms with the simple recombination algorithm. 9: Compute τ(T µ ) 10: if τ(T µ ) ≥ tol then 11: Return T and the success flag. 12: else 13: Return the mesh T

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Return the fine mesh T f obtained using the full recombination algorithm with refinement and smoothing. and 135 • and that we are allowed up to 5% triangles in the mesh. We will relax this 149 criterion in that we will measure the percentage of quadrilaterals in the mesh which 150 satisfy this criterion. Also, we will extend this into the target function which accounts 151 for the regular quadrilaterals (having the internal angles between 80 • and 100 • ) and the 152 overall percentage of the mesh covered by quadrilaterals.

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In this section we will present results of measuring the quality indicators for four 155 geometries denoted by dictionaries with co-planar plates G 1 and G 2 and two dictionaries 156 which also contain the warped plates W 1 and W 2 . The geometry G 1 is presented on 157 Figure 1, whereas the geometries G 2 , W 1 and W 2 are presented on the figures 3 and 4.

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The results will be summarized in a tables. We will also present details of some of  We see that the full recombination algorithm produces an almost fully quadrilateral mesh, but the quality of the mesh, as measured by the SIGE indicator, for the full recombination algorithm is lower. In figure (a) we have the mesh generated by pyREMAKmsh and in Figure (b) the fully recombined mesh. As a comparison we present, exemplary, the results for the mesh produced by  Table 2. Quality measures for the studied geometries. Algorithm: Full recombination algorithm with smoothing.

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Let us now consider a more involved geometry describing a superstructure of a 183 large yacht. This geometry will be denoted by G 3 and is depicted on Figures A1 and   184 A2. The generated mesh will have more than hundred thousand elements and we report 185 the quality measures for the preconditioned recombination and the full recombination 186 algorithms.
187 Table 3. The mesh quality of the superstructure of a large yacht (geometry G 3 ). The geometry, together with a detail of the performance of two algorithms on the 188 web surfaces is presented in the Figure 6. 189 We see that the pyREMAKEmsh algorithm typically achieves the quality restrictions as  Table   197 2 where the number of degenerate elements can be as high as 20%. Furthermore, the 198 SIGE criterion shows that these lower quality elements can be found in a high stress area 199 such as web surfaces. Also, on an example of a large geometry (120,000 elements for the 200 pyREMAKEmsh algorithm and almost twice as many for the full recombination algorithm) 201 we see that pyREMAKEmsh can produce the meshes with as few as less than 5 % triangles.

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Finally, let us not that the accuracy of the finite element solution for typical low 203 degree shell elements can sometimes be reduced by more than 20% when elements 204 whose internal angles are far away from the right angle are used. This is the reason 205 why we have opted to return both a fully recombined mesh as well as a preconditioned  The following abbreviations are used in this manuscript: 232 233 SIGE signed inverse gradient error for the finite element solution web surface a parallelogram with one dimension much smaller than the other regular surface quadrilateral defined by four co-planar corners warped surfaces generalized quadrilateral defined by four not co-planar corners 234 Appendix A 235 We present on Figure A1 the full mesh of the geometry G 3 as produced by the