VoroCrust: Voronoi Meshing Without Clipping

Polyhedral meshes are increasingly becoming an attractive option with particular advantages over traditional meshes for certain applications. What has been missing is a robust polyhedral meshing algorithm that can handle broad classes of domains exhibiting arbitrarily curved boundaries and sharp features. In addition, the power of primal-dual mesh pairs, exemplified by Voronoi-Delaunay meshes, has been recognized as an important ingredient in numerous formulations. The VoroCrust algorithm is the first provably-correct algorithm for conforming polyhedral Voronoi meshing for non-convex and non-manifold domains with guarantees on the quality of both surface and volume elements. A robust refinement process estimates a suitable sizing field that enables the careful placement of Voronoi seeds across the surface circumventing the need for clipping and avoiding its many drawbacks. The algorithm has the flexibility of filling the interior by either structured or random samples, while preserving all sharp features in the output mesh. We demonstrate the capabilities of the algorithm on a variety of models and compare against state-of-the-art polyhedral meshing methods based on clipped Voronoi cells establishing the clear advantage of VoroCrust output.Comment: 18 pages (including appendix), 18 figures. Version without compressed images available on https://www.dropbox.com/s/qc6sot1gaujundy/VoroCrust.pdf. Supplemental materials available on https://www.dropbox.com/s/6p72h1e2ivw6kj3/VoroCrust_supplemental_materials.pd

Advances in co-volume mesh generation and mesh optimisation techniques

This paper introduces developments in modified techniques for the generation of unstructured, non-uniform, dual orthogonal meshes which are suitable for use with co-volume solution schemes. Two new mesh generation techniques, a modified advancing front technique and an octree-Delaunay algorithm, are coupled with a mesh optimisation algorithm. When using a Delaunay–Voronoi dual, to construct mutually orthogonal meshes for co-volume schemes, it is essential to minimise the number of Delaunay elements which do not contain their Voronoi vertex. These new techniques provide an improvement over previous approaches, as they produce meshes in which the number of elements that do not contain their Voronoi vertex is reduced. In particular, it is found that the optimisation algorithm, which could be applied to any mesh cosmetics problem, is very effective, regardless of the quality of the initial mesh. This is illustrated by applying the proposed approach to a number of complex industrial aerospace geometries

A novel implementation of computational aerodynamic shape optimisation using Modified Cuckoo Search

This paper outlines a new computational aerodynamic design optimisation algorithm using a novel method of parameterising a computational mesh using `control nodes'. The shape boundary movement as well as the mesh movement is coupled to the movement of user--defined control nodes via a Delaunay Graph Mapping technique. A Modified Cuckoo Search algorithm is employed for optimisation within the prescribed design space defined by the allowed range of control node displacement. A finite volume compressible Navier--Stokes solver is used for aerodynamic modelling to predict aerodynamic design `fitness'. The resulting coupled algorithm is applied to a range of test cases in two dimensions including aerofoil lift--drag ratio optimisation intake duct optimisation under subsonic, transonic and supersonic flow conditions. The discrete (mesh--based) optimisation approach presented is demonstrated to be effective in terms of its generalised applicability and intuitiveness

Controlling First Four Moments for Robust Optimization

International audienceThe paper addresses the solution of robust moment-based optimization problems after a multipoint reformulation. The first four moments are considered (i.e. mean, variance, skewness and kurtosis) going beyond classical engineering optimization based on the control of the mean and variance. In particular, the impact on the design of a control of the third and fourth moments are discussed. The multipoint formulation leads to discrete expressions for the moments. linking moment-based and multipoint optimizations. The linearity of the sums in the discrete moments permits an easy evaluation of their gradients with respect to the design variables. Optimal sampling issues are analyzed and a procedure is proposed to quantify the confidence level on the robustness of the design. The proposed formulation is fully parallel and the time-to-solution is comparable to single-point situations. It is applied to three problems: an analytical least-square minimization problem, a shape optimization problem with a reduced-order model, and a full aircraft shape optimization robust over a range of transverse winds

Backward uncertainty propagation in shape optimization

International audienceWe aim at quantifying the impact of state uncertainties in shape optimization. This provides confidence bounds for the optimal solution. The approach is presented for inverse designs where the target is assumed uncertain. No sampling of a large dimensional space is necessary and the approach uses what is already available in a deterministic gradient-based inversion algorithm. Our proposal is based on the introduction of directional quantile-based extreme scenarios knowing the Probability Density Function (PDF) of the target data. We use these scenarios to define a matrix having the structure of the covariance matrix of the optimization parameters. We compare this construction to another one using the gradient of the functional by an adjoint method. The paper goes beyond inverse design and shows how to apply the method to general optimization problems. The ingredients of the paper are illustrated on a model problem with the Burgers equation and on the optimization of the shape of an aircraft. Overall, the computational complexity is comparable to the deterministic case

Optimal Voronoi Tessellations with Hessian-based Anisotropy

International audienceThis paper presents a variational method to generate cell complexes with local anisotropy conforming to the Hessian of any given convex function and for any given local mesh density. Our formulation builds upon approximation theory to offer an anisotropic extension of Centroidal Voronoi Tessellations which can be seen as a dual form of Optimal Delaunay Triangulation. We thus refer to the resulting anisotropic polytopal meshes as Optimal Voronoi Tessel-lations. Our approach sharply contrasts with previous anisotropic versions of Voronoi diagrams as it employs first-type Bregman diagrams , a generalization of power diagrams where sites are augmented with not only a scalar-valued weight but also a vector-valued shift. As such, our OVT meshes contain only convex cells with straight edges, and admit an embedded dual triangulation that is combinatorially-regular. We show the effectiveness of our technique using off-the-shelf computational geometry libraries

Towards an efficient computational strategy for electro-activation in cardiac mechanics

The computational modelling of the heart motion within a cardiac cycle is an extremely challenging problem due to (a) the complex multi-scale interaction that takes place between the electrophysiology and electrochemistry at cellular level and the macro-scale response of the heart muscle, and (b) the large deformations and the strongly anisotropic and quasi-incompressible behaviour of the myocardium. These pose an extreme challenge to the scalability of electro-mechanical solvers due to the size and conditioning of the system of equations required to obtain accurate solutions, both in terms of wall deformation and transmembrane potential propagation. In the search towards an efficient modelling of electro-activation, this paper presents a coupled electromechanical computational framework whereby, first, we explore the use of an efficient stabilised low order tetrahedral Finite Element methodology and compare it against a very accurate super enhanced mixed formulation previously introduced by the authors in Garcia-Blanco et al. (2019) and, second, we exploit the use of tailor-made staggered and staggered linearised solvers in order to assess their feasibility against a fully monolithic approach. Through a comprehensive set of examples, culminating in a realistic ventricular geometry, we aim to put forward some suggestions regarding the level of discretisation and coupling required to ensure sufficiently reliable results yet with an affordable computational time