New Challenges in Grid Generation and Adaptivity for Scientific Computing

Implicit boundary means that the boundaries and/or interfaces between domains are not anymore defined by an explicit boundary mesh but rather by an implicit function. It is the case with embedded boundary methods or immersed boundary methods. Here we consider a filtered level set methods and meshing is then performed using an anisotropic mesh adaptation framework applied to the level sel interpolation. The interpolation error estimate is driving the adaptive process giving rise to a new way of boundary recovery. The accuracy of the recovery process depends then on the user given parameter, an arbitrary thickness of the interface. The thickness is normally related to the mesh size, but it is shown that adaptive meshing enables to reverse this condition: fixing the thickness parameter and accounting for the adaptation process to fulfill the mesh size condition. Several examples are given to demonstrate the potential of this approach.

We present a method for re-meshing surfaces in order to follow the intrinsic anisotropy of the surfaces. In particular, we use the information related to the normals to the surfaces, and embed the surfaces into a higher dimensional space (here we embed the surfaces in a six-dimensional space). This allows us to settle an isotropic mesh optimization problem in this embedded space: starting from an initial mesh of a surface, we optimize the mesh by improving the mesh quality measured in the embedded space. The mesh is optimized by properly combining common local mesh operations, i.e., edge flipping, edge contraction, vertex smoothing, and vertex insertion. All operations are applied directly on the three-dimensional surface mesh and the resulting mesh is curvature adapted. This new method improves the approach proposed by Lévy and Bonneel (Variational anisotropic surface meshing with Voronoi parallel linear enumeration. In: Proceedings of the 21st International Meshing Roundtable, pp. 349–366. Springer, New York, 2013), by allowing to preserve sharp features. The reliability and robustness of the proposed re-meshing technique is shown via a number of examples.

We develop a reliable a posteriori anisotropic first order estimator for the numerical simulation of the Frankfort and Marigo model of brittle fracture, after its approximation by means of the Ambrosio-Tortorelli variational model. We show that an adaptive algorithm based on this estimator reproduces all the previously obtained well-known benchmarks on fracture development with particular attention to the fracture directionality. Additionally, we explain why our method, based on an extremely careful tuning of the anisotropic adaptation, has the potential of outperforming significantly in terms of numerical complexity the ones used to achieve similar degrees of accuracy in previous studies.

We study the problem of maintaining a deforming surface mesh, specified only by a dense sample of n points that move with the surface. We propose a motion model under which the class of \((\varepsilon,\alpha )\)-meshes can be efficiently maintained by a combination of edge flips and insertion and deletion of vertices. We can enforce bounded aspect ratios and a small approximation error throughout the deformation.

We present a new strategy, based on the idea of the meccano method and a novel T-mesh optimization procedure, to construct a T-spline parameterization of 2D geometries for the application of isogeometric analysis. The proposed method only demands a boundary representation of the geometry as input data. The algorithm obtains, as a result, high quality parametric transformation between 2D objects and the parametric domain, the unit square. First, we define a parametric mapping between the input boundary of the object and the boundary of the parametric domain. Then, we build a T-mesh adapted to the geometric singularities of the domain in order to preserve the features of the object boundary with a desired tolerance. The key of the method lies in defining an isomorphic transformation between the parametric and physical T-mesh finding the optimal position of the interior nodes by applying a new T-mesh untangling and smoothing procedure. Bivariate T-spline representation is calculated by imposing the interpolation conditions on points sited both on the interior and on the boundary of the geometry. Proposed method also permits the modeling of objects with embedded geometries that can be used to solve problems with domains composed of several materials. Application of the isogeometric analysis to these type of domains are presented. The effectiveness of the proposed technique is shown in several examples.

Mesh adaptation is a powerful way to minimise the computational cost of mesh based computation. It is particularly successful for multi-scale problems where the required mesh resolution can vary by orders of magnitude across the domain. The end result is local control over solution accuracy and reduced time to solution.

In the case of large scale simulations, where the time to solution is unacceptable or the memory requirements exceeds available RAM, mesh based computation is typically parallelised using domain decomposition methods using the Message Passing Interface (MPI). This allows a simulation to run in parallel on a distributed memory computer. While this has been a high successful strategy up until now, the drive towards low power multi- and many-core architectures means that an even higher degree of parallelism is required and the memory hierarchy exploited to maximise memory bandwidth.

For this reason application codes are increasingly adopting a hybrid parallel approach whereby decomposition methods, implemented using the Message Passing Interface (MPI), are applied for inter-node parallelisation, while a threaded programming model is used for intra-node parallelisation. Mesh adaptivity has been successfully parallelised using MPI by a number of groups, and can be implemented efficiently with few modifications to the serial code. However, thread-level parallelism is significantly more challenging because each thread modifies the mesh data and therefore must be carefully marshalled to avoid data races while still ensuring enough parallelism is exposed to achieve good parallel efficiency.

Here we describe a new thread-parallel algorithm for anisotropic mesh adaptation algorithms. For each mesh optimisation phase (refinement, coarsening, swapping and smoothing) we describe how independent sets of tasks are defined. We show how a deferred updates strategy can be used to update the mesh data structures in parallel and without data contention. We show that despite the complex nature of mesh adaptation and inherent load imbalances in the mesh adaptivity, good parallel efficiency can be achieved.

We present a new immersed method for Computational Fluid Dynamics applications. It is based on the use of Non Uniform Rational B-Splines (NURBS). The distance function to an immersed solid is computed directly from its Computer Aided Design (CAD) description. This allows to bypass the generation of surface meshes and to obtain accurate levelset functions for complex geometries. Combined with a metric based anisotropic mesh adaptation and stabilized Finite Elements Method (FEM), it allows a novel, efficient and flexible approach to deal with a wide range of fluid structure interaction problems. The metric field is computed directly at the node of the mesh using the length distribution tensor and an edge based error analysis. Several 2D and 3D numerical examples will demonstrate the applicability of the proposed method.

This paper outlines some recent developments in the process of generating well centered tetrahedral meshes. A well centered tetrahedron contains its circumcentre, which is a basic property required for a valid co-volume discretisation. Although most work in this area has focussed on improving meshes generated using classical techniques, in this paper we consider modification of the generation procedure itself. A simple lattice point insertion technique is introduced and the potential of the technique for generating well centered meshes is demonstrated. This is accomplished by comparing, for some complex geometries, the meshes generated with the meshes created by a standard Delaunay mesh refinement technique. Despite the simplicity of the lattice point insertion method, the comparison is found to be favourable and the method is shown to produce good well centered elements in the vicinity of the geometry.

In this work we describe a unified approach that blends the best characteristics of both a near body pseudo-structured boundary-layer (BL) and generalized anisotropic metric approaches. Specifically, near-body physics with anisotropy are resolved using an a priori pseudo-structured process and off-body or field features are resolved using an adaptive generalized approach. In particular the metric field of the adaptive approach is derived from the BL region of the pseudo-structured approach. The derived metric is based on local aspect ratio and geometry. This metric is then blended from the BL region into the overall field to allow for a smooth transition to the generalized field. The result is a flexible and optimal overall mesh generation process that can be used with or without adaptation. Metric-based formulations for quality functions and other geometric quantities require for mesh generation are presented. Results are presented that demonstrate the overall approach in the context of blending between the near body pseudo-structured region and the outer tetrahedral field region. These results point out that the metric-based transition can be used to improve mesh quality and density for configurations with anisotropic surface meshes and BL regions that do not reach outer region length scale.

Recently, a new mesh generation technique based on the isoparametric representation of curvilinear elements has been developed in order to address the issue of generating high-order meshes with highly stretched elements. Given a valid coarse mesh comprising of a prismatic boundary layer, this technique uses the shape functions that define the geometries of the elements to produce a series of subdivided elements of arbitrary height. The purpose of this article is to investigate the range of conditions under which the resulting meshes are valid, and additionally to consider the application of this method to different element types. We consider the subdivision strategies that can be achieved with this technique and apply it to the generation of meshes suitable for boundary-layer fluid problems.

An adaptive, anisotropic finite element algorithm is proposed to solve the 3D linear elasticity equations in a thin 3D plate. Numerical experiments show that adaptive computations can be performed in thin 3D domains having geometrical aspect ratio 1:1000.

We assess the impact of space-time mesh adaptivity on the modeling of solute transport in porous media. This approach allows an automatic selection of both the spatial mesh and the time step on the basis of a suitable recovery-based error estimator. In particular, we deal with an anisotropic control of the spatial mesh. The solver coupled with the adaptive module deals with an advection-dispersion equation to model the transport of dissolved species, which are assumed to be convected by a Darcy flow field. The whole solution-adaptation procedure is assessed through two-dimensional numerical tests. A numerical convergence analysis of the spatial mesh adaptivity is first performed by considering a test-case with analytical solution. Then, we validate the space-time adaptive procedure by reproducing a set of experimental observations associated with solute transport in a homogeneous sand pack. The accuracy and the efficiency of the methodology are discussed and numerical results are compared with those associated with fixed uniform space-time discretizations. This assessment shows that the proposed approach is robust and reliable. In particular, it allows us to obtain a significant improvement of the simulation quality of the early solute arrivals times at the outlet of the medium.

We propose a framework for performing anisotropic mesh deformations. Our goal is to produce high quality meshes with no inverted elements on domains which undergo large deformations. To the greatest extent possible, the meshes should have similar element shape; however, topological changes are performed as necessary in order to improve mesh quality. Our framework is based upon the previous work of two of the authors and their collaborators (Kim et al., Int. J. Numer. Methods Eng. 94(1):20–42, 2013; Kim et al., Computer and Mathematics with Applications, Submitted, November 2014) and consists of four steps. The first step is to perform anisotropic finite element-based mesh warping to estimate the interior vertex positions based upon an appropriate choice of the PDE coefficients. The second step is to perform multiobjective mesh optimization in order to eliminate inverted elements and improve element shape. Edge swaps are then performed to further improve the mesh quality. A final mesh smoothing pass is then performed. Our numerical results show that our framework can be used to generate high quality meshes with no inverted elements for very large deformations. In particular, the addition of topological changes to our hybrid mesh deformation algorithm (Kim et al., Computer and Mathematics with Applications, Submitted, November 2014) proved to be an extremely efficient way of improving the mesh quality.

We present an in-depth analysis and benchmark of shape deformation techniques for their use in simulation-based design optimization scenarios. We first introduce classical free-form deformation, its direct manipulation variant, as well as deformations based on radial basis functions. We compare the techniques in a series of representative synthetic benchmarks, including computational performance, numerical robustness, quality of the deformation, adaptive refinement, as well as precision of constraint satisfaction. As an application-oriented benchmark we investigate the ability to adapt an existing volumetric simulation mesh according to an updated surface geometry, including unstructured tetrahedral, structured hexahedral, and arbitrary polyhedral example meshes. Finally, we provide a detailed assessment of the methods and give concrete advice on choosing a suitable technique for a given optimization scenario.

The objective of this work is to create grids for free-surface water flow simulation entirely with automatic grid refinement. It is shown why it is necessary to refine the mesh iteratively as the solution converges and why refinement and derefinement of hexahedral cells must be treated anisotropically.The proposed refinement criterion is a combination of the pressure Hessian with refinement at the free surface, in order to capture the flow which drives the surface motion and the position of the surface itself. Smoothing is needed in the computation of the Hessian in order to remove oscillations in the pressure, the pressure Hessian is extrapolated through the free surface to remove its discontinuity there.Two test cases confirm that effective fine meshes for wave computation can be created with the proposed automatic refinement procedure.