A distortion measure to validate and generate curved high-order meshes on CAD surfaces with independence of parameterization

This is the accepted version of the following article: [Gargallo-Peiró, A., Roca, X., Peraire, J., and Sarrate, J. (2016) A distortion measure to validate and generate curved high-order meshes on CAD surfaces with independence of parameterization. Int. J. Numer. Meth. Engng, 106: 1100–1130. doi: 10.1002/nme.5162], which has been published in final form at http://onlinelibrary.wiley.com/doi/10.1002/nme.5162/abstractA framework to validate and generate curved nodal high-order meshes on Computer-Aided Design (CAD) surfaces is presented. The proposed framework is of major interest to generate meshes suitable for thin-shell and 3D finite element analysis with unstructured high-order methods. First, we define a distortion (quality) measure for high-order meshes on parameterized surfaces that we prove to be independent of the surface parameterization. Second, we derive a smoothing and untangling procedure based on the minimization of a regularization of the proposed distortion measure. The minimization is performed in terms of the parametric coordinates of the nodes to enforce that the nodes slide on the surfaces. Moreover, the proposed algorithm repairs invalid curved meshes (untangling), deals with arbitrary polynomial degrees (high-order), and handles with low-quality CAD parameterizations (independence of parameterization). Third, we use the optimization procedure to generate curved nodal high-order surface meshes by means of an a posteriori approach. Given a linear mesh, we increase the polynomial degree of the elements, curve them to match the geometry, and optimize the location of the nodes to ensure mesh validity. Finally, we present several examples to demonstrate the features of the optimization procedure, and to illustrate the surface mesh generation process.Peer ReviewedPostprint (author's final draft

Improving the quality of finite volume meshes through genetic optimisation

Author's accepted version. The final publication is available at Springer via http://dx.doi.org/10.1007/s00366-015-0423-0Mesh quality issues can have a substantial impact on the solution process in Computational Fluid Dynamics (CFD), leading to poor quality solutions, hindering convergence and in some cases, causing the solution to diverge. In many areas of application, there is an interest in automated generation of finite volume meshes, where a meshing algorithm controlled by pre- specified parameters is applied to a pre-existing CAD geometry. In such cases the user is typically confronted with a large number of controllable parameters, and ad- justing these takes time and perserverence. The process can however be regarded as a multi-input and possi- bly multi-objective optimisation process which can be optimised by application of Genetic Algorithm tech- niques. We have developed a GA optimisation code in the language Python, including an implementation of the NGSA-II multi-objective optimisation algorithm, and applied to control the mesh generation process us- ing the snappyHexMesh automated mesher in Open- FOAM. We demonstrate the results on three selected cases, demonstrating significant improvement in mesh quality in all cases

Validation and generation of curved meshes for high-order unstructured methods

In this thesis, a new framework to validate and generate curved high-order meshes for complex models is proposed. The main application of the proposed framework is to generate curved meshes that are suitable for finite element analysis with unstructured high-order methods. Note that the lack of a robust and automatic curved mesh generator is one of the main issues that has hampered the adoption of high-order methods in industry. Specifically, without curved high-order meshes composed by valid elements and that match the domain boundary, the convergence rates and accuracy of high-order methods cannot be realized. The main motivation of this work is to propose a framework to address this issue. First, we propose a definition of distortion (quality) measure for curved meshes of any polynomial degree. The presented measures allow validating if a high-order mesh is suitable to perform finite element analysis with an unstructured high-order method. In particular, given a high-order element, the measures assign zero quality if the element is invalid, and one if the element corresponds to the selected ideal configuration (desired shape and nodal distribution). Moreover, we prove that if the quality of an element is not zero, the region where the determinant of the Jacobian is not positive has measure zero. We present several examples to illustrate that the proposed measures can be used to validate high-order isotropic and boundary layer meshes. Second, we develop a smoothing and untangling procedure to improve the quality for curved high-order meshes. Specifically, we propose a global non-linear least squares minimization of the defined distortion measures. The distortion is regularized to allow untangling invalid meshes, and it ensures that if the initial configuration is valid, it never becomes invalid. Moreover, the optimization procedure preserves, whenever is possible, some geometrical features of the linear mesh such as the shape, stretching, straight-sided edges, and element size. We demonstrate through examples that the implementation of the optimization problem is robust and capable of handling situations in which the mesh before optimization contains a large number of invalid elements. We consider cases with polynomial approximations up to degree ten, large deformations of the curved boundaries, concave boundaries, and highly stretched boundary layer elements. Third, we extend the definition of distortion and quality measures to curved high-order meshes with the nodes on parameterized surfaces. Using this definition, we also propose a smoothing and untangling procedure for meshes on CAD surfaces. This procedure is posed in terms of the parametric coordinates of the mesh nodes to enforce that the nodes are on the CAD geometry. In addition, we prove that the procedure is independent of the surface parameterization. Thus, it can optimize meshes on CAD surfaces defined by low-quality parameterizations. Finally, we propose a new mesh generation procedure by means of an a posteriori approach. The approach consists of modifying an initial linear mesh by first, introducing high-order nodes, second, displacing the boundary nodes to ensure that they are on the CAD surface, and third, smoothing and untangling the resulting mesh to produce a valid curved high-order mesh. To conclude, we include several examples to demonstrate that the generated meshes are suitable to perform finite element analysis with unstructured high-order methods.Postprint (published version

Unstructured Grid Adaptation: Status, Potential Impacts, and Recommended Investments Towards CFD 2030

International audienceUnstructured grid adaptation is a powerful tool to control Computational Fluid Dynamics (CFD) discretization error. It has enabled key increases in the accuracy, automation, and capacity of some fluid simulation applications. Slotnick et al. provide a number of case studies in the CFD Vision 2030 Study: A Path to Revolutionary Computational Aerosciences to illustrate the current state of CFD capability and capacity. The study authors forecast the potential impact of emerging High Performance Computing (HPC) environments forecast in the year 2030 and identify that mesh generation and adaptivity will continue to be significant bottlenecks in the CFD workflow. These bottlenecks may persist because very little government investment has been targeted in these areas. To motivate investment, the impacts of improved grid adaptation technologies are identified. The CFD Vision 2030 Study roadmap and anticipated capabilities in complementary disciplines are quoted to provide context for the progress made in grid adaptation in the past fifteen years, current status, and a forecast for the next fifteen years with recommended investments. These investments are specific to mesh adaptation and impact other aspects of the CFD process. Finally, a strategy is identified to di↵use grid adaptation technology into production CFD work flows

The optimisation of finite element meshes

Among the several numerical methods which are available for solving complex problems in many areas of engineering and science such as structural analysis, fluid flow and bio-mechanics, the Finite Element Method (FEM) is the most prominent. In the context of these methods, high quality meshes can be crucial to obtaining accurate results. Finite Element meshes are composed of elements and the quality of an element can be described as a numerical measure which estimates the effect that the size/shape of an element will have on the accuracy of an analysis. In this thesis, the strong link between mesh geometry and the accuracy and efficiency of a simulation is explored and it is shown that poor quality elements cause both interpolation errors and poor conditioning of the global stiffness matrix. Numerical optimisation is the process of maximising or minimising an objective func- tion, subject to constraints on the solution. When this is applied to a finite element mesh it is referred to as mesh optimisation, where the quality of the mesh is the objec- tive function and the constraints include, for example, the domain geometry, maximum element size, etc. A mesh optimisation strategy is developed with a particular focus on optimising the quality of the worst elements in a mesh. Using both two and three dimensional examples, the most efficient and effective combination of element quality measure and objective function is found. Many of the problems under consideration are characterised by very complex geometries. The nodes lying on the surfaces of such meshes are typically treated as unmovable by most mesh optimisation software. Techniques exist for moving such nodes as part of the mesh optimisation process, however, the resulting mesh geometry and area/volume is often not conserved. This means that the optimised mesh is no longer an accurate discretisation of the original domain. Therefore, a method is developed and demonstrated which optimises the positions of surface nodes while respecting the geometry and area/volume of a domain. At the heart of many of the problems being considered is the Arbitrary Lagrangian Eulerian (ALE) formulation where the need to ensure mesh quality in an evolving mesh is very important. In such a formulation, a method of determining the updated nodal positions is required. Such a method is developed using mesh optimisation techniques as part of the FE solution process and this is demonstrated using a two-dimensional, axisymmetric simulation of a micro-fluid droplet subject to external excitation. While better quality meshes were observed using this method, the time step collapsed resulting in simulations requiring significantly more time to complete. The extension of this method to incorporate adaptive re-meshing is also discussed

Survey on Additive Manufacturing, Cloud 3D Printing and Services

Cloud Manufacturing (CM) is the concept of using manufacturing resources in a service oriented way over the Internet. Recent developments in Additive Manufacturing (AM) are making it possible to utilise resources ad-hoc as replacement for traditional manufacturing resources in case of spontaneous problems in the established manufacturing processes. In order to be of use in these scenarios the AM resources must adhere to a strict principle of transparency and service composition in adherence to the Cloud Computing (CC) paradigm. With this review we provide an overview over CM, AM and relevant domains as well as present the historical development of scientific research in these fields, starting from 2002. Part of this work is also a meta-review on the domain to further detail its development and structure