A unified approach for a posteriori highorder cur

Authors: Roman Poya Ruben Sevilla Antonio J Gil

Publish Date: 2016/06/06

Volume: 58, Issue: 3, Pages: 457-490

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Abstract

The paper presents a unified approach for the a posteriori generation of arbitrary highorder curvilinear meshes via a solid mechanics analogy The approach encompasses a variety of methodologies ranging from the popular incremental linear elastic approach to very sophisticated nonlinear elasticity In addition an intermediate consistent incrementally linearised approach is also presented and applied for the first time in this context Utilising a consistent derivation from energy principles a theoretical comparison of the various approaches is presented which enables a detailed discussion regarding the material characterisation calibration employed for the different solid mechanics formulations Five independent quality measures are proposed and their relations with existing quality indicators used in the context of a posteriori mesh generation are discussed Finally a comprehensive range of numerical examples both in two and three dimensions including challenging geometries of interest to the solids fluids and electromagnetics communities are shown in order to illustrate and thoroughly compare the performance of the different methodologies This comparison considers the influence of material parameters and number of load increments on the quality of the generated highorder mesh overall computational cost and crucially the approximation properties of the resulting mesh when considering an isoparametric finite element formulationThe performance of highorder discretisation methods for the simulation of various problems in science and engineering has been the object of intensive research during the last two decades 32 41 44 67 These methods have the potential to offer an increased level of accuracy with a reduced number of degrees of freedom and more importantly a reduced computational cost 16 35 63The potential of highorder unstructured methods has been intensively studied by the computational fluid dynamics CFD community in the last decade due to their inherent ability to accurately predict the behaviour of complex high Reynolds number flows 37 45 48 74 It is also well known that loworder methods are highly dissipative and extremely refined meshes are required to properly resolve the propagation of vortices over long distances The advantages of highorder methods have also attracted the attention of researchers working in wave propagation problems eg acoustics and electromagnetics due to their low dispersion and dissipation compared to loworder methods 2 7 31 46 47 64 In particular the highorder discontinuous Galerkin method has become popular in this area due to its ability to propagate waves over long periods of time with a reduced computational cost compared to alternative loworder methods 13 15 38 40 42The use of curved elements is nowadays accepted to be crucial in order to fully exploit the advantages of highorder discretisation methods 5 19 43 49 61 62 70 78 but until relatively recently the challenge of automatically generating highorder curvilinear meshes has been an obstacle for the widespread application of highorder methods 73 Methods to produce highorder curvilinear meshes are traditionally classified into direct methods and a posteriori methods 20 21 Direct methods build the curvilinear highorder mesh directly from the CAD boundary representation of the domain whereas a posteriori approaches rely on mature loworder mesh generation algorithms to produce an initial mesh that is subsequently curved using different techniques such as local modification of geometric entities 20 21 50 65 66 solid mechanics analogies 58 77 or optimisation 26 71Within the category of a posteriori approaches the solid mechanics analogy first proposed in 58 has become increasingly popular The main idea is to consider the initial loworder mesh as the undeformed configuration of an elastic solid Highorder nodal distributions are then inserted into all of the elements and then the nodes over element edges/faces in contact with the curved parts of the boundary are projected onto the true CAD boundary The displacement required to move the nodes onto the true boundary is interpreted as an essential boundary condition within the solid mechanics analogy The solution of the elastic problem provides the desired curvilinear mesh as the deformed configuration The initial approach proposed in 58 used a nonlinear neoHookean constitutive model Several attempts to reduce the computational cost of this approach have been proposed based on a linear elastic analogy see 1 77 It is clear that when large deformations are induced to produce the deformed curvilinear highorder mesh a linear elastic model can result in nonvalid elements due to the violation of the hypothesis of small deformations In order to alleviate this problem it is possible to split the desired potentially large displacement of boundary nodes into smaller load increments Other approaches to increase the robustness of the linear elastic analogy have been recently introduced see for instance 56 where pseudo thermal effects are introduced It is worth noting that mesh moving strategies based on an elastic analogy have also been proposed and successfully used with a proven track record of robustness in the loworder context 39 68 69Although some approaches have been demonstrated to be capable of producing curvilinear highorder meshes of highly complex geometrical configurations including anisotropic boundary layer meshes around a full aircraft configuration 77 a comparison of the proposed solid mechanics analogies has not been investigated With this comparison in mind in this paper a unified theoretical and computational solid mechanics approach is proposed This formulation encompasses the linear and nonlinear formulations proposed in 58 and 77 respectively In addition a new incrementally linearised elasticity formulation not previously applied to generate curvilinear highorder meshes is proposed within this unified approach Different distortion measures are considered in order to evaluate the quality of the generated meshes for the different formulations analysing the effect of material parameters load increments computational cost and more importantly the approximation properties of the resulting highorder meshThe paper is organised as follows In Sect 2 the fundamentals of nonlinear continuum mechanics are briefly revisited by following some recent developments in 11 12 where the kinematics of the nonlinear continua and the principle of virtual work for a displacementbased formulation in material and spatial settings are presented The new consistent incrementally linearised approach is detailed in Sect 3 and the material characterisation for all the different formulations is described in detail in Sect 4 Using the derivation of all the formulations from an energy principle a range of quality measures are proposed in Sect 5 and their relations with existing quality indicators is briefly discussed Finally Sect 6 presents a number of numerical examples both in two and three dimensions and an extensive comparison of performances of the different formulations is presented The examples include geometries appearing in a range of areas of computational mechanics eg computational solid mechanics CFD and computational eletromagnetics Meshes are produced for a variety of degrees of approximation and for interior and exterior domains illustrating the potential of the proposed approach

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