A unified approach for a posteriori highorder curved mesh generation using solid mechanics
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 formulation.
Keywords
Highorder mesh generation Solid mechanics analogy Curved elements Isoparametric finite elements1 Introduction
The 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, 63].
The 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 (e.g. 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, 42].
The 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, 71].
Within 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, 69].
Although 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 mesh.
The 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, e.g. 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.
2 Nonlinear continuum mechanics
2.1 Kinematics
2.2 The principle of virtual work in material and spatial settings
While a myriad of methodologies can be applied to solve for the deformation of a continuum described by the motion map \(\varvec{\phi }\), such as optimisation and projection techniques [20, 21, 25, 28, 50, 51, 57, 65, 66], in the context of solid mechanics, the deformation of a continuum from its undeformed configuration to its deformed configuration can be posed as a problem of minimisation of the total potential energy \(\varPi \), subjected to certain desired constraints [9, 34, 52]. In other words, the displacement of a deformable body can be obtained by finding the stationary condition of the total potential energy, also called the principle of virtual work (or variational principle), of an assumed strain energy density.
3 A consistent incrementally linearised solid mechanics approach
Computational requirement of different solid mechanics formulations for curved mesh generation
Formulation\computational requirement  Requires increments  Requires iteration  Accounts for stresses  Tangent operator evaluation 

Nonlinear elasticity  ✓  ✓  ✓  Per iteration 
Consistent Incrementally Linearised (CIL)  ✓  ✗  ✓  Per increment 
Inconsistent Incrementally Linearised  ✓  ✗  ✓  Once at the origin 
Incremental Linear Elasticity (ILE)  ✓  ✗  ✗  Once at the origin 
Classical linear elasticity  ✗  ✗  ✗  Once at the origin 
 1.
When the state of deformation at increment \(n+1\) is obtained by computing the residual stresses, the constitutive stiffness and the geometric stiffness at the previous deformed configuration i.e. based on \(\{\mathcal {R}_n,\mathcal {C}_n,\mathcal {G}_n\}\). This consistent incrementally linearised methodology for highorder curved mesh generation, is presented in this paper for the first time.
 2.
When the residual stresses at increment \(n+1\) are obtained from the previous deformed configuration based on \(\mathcal {R}_n\), but the constitutive stiffness is evaluated at the initial undeformed (or stressfree) configuration i.e. \(\mathcal {C}_0\) and the geometric stiffness term is absent from the formulation. The technique developed by [56] falls into this category.
 3.
When both the residual stresses and constitutive stiffness at increment \(n+1\) are computed based on the initial undeformed configuration i.e. \(\mathcal {R}_0\) and \(\mathcal {C}_0\) and the geometric stiffness term is absent from the formulation, but the geometry itself is updated incrementally such that \(\varvec{x}_{n+1}=\varvec{x}_n + \varvec{u}\). This approach has been pursued in [1, 77] and from here onwards we will refer to this approach as the incremental linear elastic approach.
 4.
When \(n=0\) or in other words, when the residual stresses, constitutive stiffness and geometric stiffness are evaluated once at the initial undeformed configuration \(\{\mathcal {R}_0,\mathcal {C}_0,\mathcal {G}_0\}\) i.e. particularisation to the case of classical linear elasticity.
4 Hyperelasticity and material characterisation for different solid mechanics formulations
Alternative strain energies and their corresponding stresses, tangent operator and calibrated material parameters
Alternative internal energies  
Compressible Mooney–Rivlin  
\(\varPsi (\varvec{C})\)  \(\alpha I_1 + \beta I_2  4\beta \,\sqrt{I_3}  2\alpha \text {ln}\sqrt{I_3} + \frac{\lambda }{2}(\sqrt{I_3}1)^2 (3\alpha \beta )\) 
\(\varvec{\sigma }\)  
\([\varvec{c}]_{ijkl}\)  \(\frac{2\beta }{J}\bigg (2\,[\varvec{b}]_{ij}[\varvec{b}]_{kl}  [\varvec{b}]_{ik}[\varvec{b}]_{jl}  [\varvec{b}]_{il}[\varvec{b}]_{jk} \bigg ) + \bigg (\lambda (2J1)4\beta \bigg )\delta _{ij}\delta _{kl}  \left( \lambda (J1) 4\beta \frac{2\alpha }{J}\right) \bigg (\delta _{ik}\delta _{jl} + \delta _{il}\delta _{jk}\bigg )\) 
\(\leftrightarrow \)  \(\lambda = \lambda _{\text {lin}}\), \(\alpha +\beta = \frac{\mu _{\text {lin}}}{2}^{\mathrm{a}}\) 
Nearly incompressible Mooney–Rivlin (NIMR)  
\(\varPsi (\varvec{C})\)  \(\alpha \,{I_3}^{1/3}I_1 + \beta \,{I_3}^{1}{I_2}^{3/2}+\frac{\kappa }{2}(\sqrt{I_3}1)^2  (3\alpha +3\sqrt{3}\beta )\) 
\(\varvec{\sigma }\)  \(2\alpha \,J^{5/3}\varvec{b} +\bigg ( \kappa (J1)  \frac{2\alpha }{3}J^{5/3}\text {tr}\varvec{b} + \beta \,J^{3}\text {tr}\varvec{g}^{3/2} \bigg )\varvec{I} 3\beta \,J^{3}\text {tr}\varvec{g}^{1/2}\varvec{g}\) 
\([\varvec{c}]_{ijkl}\)  \(\frac{4\alpha }{3}J^{5/3} \bigg ([\varvec{b}]_{ij}\delta _{kl} + \delta _{ij}[\varvec{b}]_{kl}\bigg ) + \bigg (\frac{4\alpha }{9}J^{5/3}[\varvec{b}]_{mm} + \beta J^{3} ([\varvec{g}]_{mm})^{3/2} + \kappa (2\,J1) \bigg )\delta _{ij}\delta _{kl} + \bigg ( \frac{2\alpha }{3}J^{5/3}[\varvec{b}]_{mm}  \beta J^{3} ([\varvec{g}]_{mm})^{3/2}  \kappa (J1) \bigg ) \bigg (\delta _{ik}\delta _{jl} + \delta _{il}\delta _{jk}\bigg ) 3\,\beta J^{3} ([\varvec{g}]_{mm})^{1/2} \bigg (\delta _{ij}[\varvec{g}]_{kl} + [\varvec{g}]_{ij}\delta _{kl} \bigg ) +6\,\beta J^{3} ([\varvec{g}]_{mm})^{1/2} \bigg (\delta _{ik}[\varvec{g}]_{jl} + [\varvec{g}]_{il}\delta _{jk} \bigg ) +3\,\beta J^{3} ([\varvec{g}]_{mm})^{1/2} [\varvec{g}]_{ij}[\varvec{g}]_{kl}\) 
\(\leftrightarrow \)  \(2\alpha +3\sqrt{3}\beta = \mu _{\text {lin}}\), \(\kappa  \frac{4}{3}\alpha  2\sqrt{3}\beta = \lambda _{\text {lin}}^{\mathrm{b}}\) 
Transversely Isotropic Hyperelastic (TI)  
\(\varPsi (\varvec{C})\)  \( \alpha (I_13) + \beta (I_23)  \tilde{\mu }\text {ln}\sqrt{I_3} + \frac{\lambda }{2}(\sqrt{I_3}1)^2 +\eta _1(I_41)+\eta _2(I_13)(I_41)+\gamma (I_41)^2 \frac{\eta _1}{2}(I_5  1)\) 
\(\varvec{\sigma }\)  
\([\varvec{c}]_{ijkl}\)  \(\frac{2\beta }{J}\bigg (2\,[\varvec{b}]_{ij}[\varvec{b}]_{kl}  [\varvec{b}]_{ik}[\varvec{b}]_{jl}  [\varvec{b}]_{il}[\varvec{b}]_{jk} \bigg ) + \lambda (2J1)\delta _{ij}\delta _{kl} +\bigg (\tilde{\mu }  \lambda (J1)\bigg ) \bigg (\delta _{ik}\delta _{jl} + \delta _{il}\delta _{jk}\bigg ) + \frac{4\eta _2}{J}([\varvec{b}]_{ij}[\varvec{FN}]_k[\varvec{FN}]_l +[\varvec{FN}]_i[\varvec{FN}]_j[\varvec{b}]_{kl}) +\frac{8\gamma }{J} [\varvec{FN}]_i[\varvec{FN}]_j[\varvec{FN}]_k[\varvec{FN}]_l \frac{2\eta _1}{J}\bigg ([\varvec{b}]_{jk}[\varvec{FN}]_i[\varvec{FN}]_l + [\varvec{b}]_{ik}[\varvec{FN}]_j[\varvec{FN}]_l \bigg ) \) 
\(\leftrightarrow \)  \(\tilde{\mu } = 2\alpha +4\beta \), \(\alpha +\beta =\frac{E}{4(1+\nu )}\), \(\eta _1 = 4\alpha  G_A\), \(12\alpha +\lambda =C_{11}\), \(4\eta _2=C_{13}  4\alpha  \lambda \), \(8\gamma =C_{33}  12\alpha +4\eta _18\eta _2\lambda ^{\mathrm{c}}\) 
4.1 The nonlinear hyperelastic case
4.2 The classical linear elastic case
5 Mesh quality measures
Quality or distortion measures are traditionally used both in a low and highorder finite element context in order to quantify the approximation properties induced by a computational mesh. In a standard highorder finite element formulation, measures involving the Jacobian of the isoparametric mapping have been extensively used [29, 77], in particular the socalled scaled Jacobian. This measure only quantifies volumetric deformations and alternative measures that exploit different modes of deformation and account for shape, skewness and degeneracy of elements have only been recently considered [27, 28]. However, it is worth noting that not all of these quality measures can be regarded as independent quantities.
In practice, the invariants are evaluated at a discrete set of points within the reference element, usually the quadrature points that will be employed during a computational simulation. For the numerical examples presented here, a quadrature rule is used that integrates polynomials of degree up to 2p, where p is the order of approximation.
In order to obtain a representative quality measure for a given computational mesh, a variety of statistical data can be reported, such as the mean quality or the standard deviation. However, in the numerical examples presented here, the mesh quality is defined by computing the minimum over all the elements, namely \(Q_j=\min _{e}\;\{Q_j^e \}\). Despite this being the least favourable choice, it is well known that a few low quality elements can substantially deteriorate the overall quality of a finite element simulation, specially if these elements are near a curved boundary. Several numerical examples in two and three dimensions are used in the next section to evaluate the performance of different approaches for a posteriori mesh generation. The objective is to produce meshes where the minimum quality is as high as possible as this will provide the better approximation properties of a highorder finite element solver.
From the mesh distortion point of view, the first quality measure \(Q_1\), quantifies fibre deformation (for instance, distortion of the edges of an element), the second quality measure \(Q_2\), quantifies surface deformation (for instance, distortion of the faces of an element) and the third quality measure \(Q_3\), quantifies volumetric deformation (distortion of the element itself). In fact, it is worth noting that the scaled Jacobian corresponds to the quality measure \(Q_3\). For simplicial elements (i.e. triangles and tetrahedra), this measure is identical to the Jacobian of the deformation gradient tensor J because the isoparametric mapping for a simplicial elements with planar faces (or edges in two dimensions) is affine. This result is valid because, in the context of a posteriori highorder mesh generation, the undeformed configuration typically corresponds to a mesh formed by elements with planar faces (or edges).
The quality measures \(Q_4\) and \(Q_5\), based on the two anisotropic invariants, quantify the distortion in the direction of anisotropy. These measures can only be utilised when the internal energy of the material is anisotropic and, since in the context of curved mesh generation this is not often the case, their usage remains limited. Moreover, anisotropic quality measures are typically dependent on the geometrical parametrisation.
Finally, it should be emphasised that, in contrast to the nonlinear approach, the solution of the incrementally linearised problem in (13), does not correspond to the minimisation of the total potential energy (11) with respect to the fundamental strain measures \(\{\varvec{F},\varvec{H},J\}\) per se, but rather with respect to the incrementally linearised versions of these quantities. Furthermore, it is easy to identify that for plane strain problems, the first and the second invariants are indeed identical i.e. \(I_1=I_2\) which in turn translates into the corresponding quality measures being identical \(Q_1=Q_2\). This is only true for twodimensional plane strain problems.
6 Examples
This section presents a detailed comparison of the various solid mechanics formulations considered in this work (refer to Table 1) for the a posteriori generation of highorder curvilinear meshes. The comparison focuses on the advantages and disadvantages of the various formulations, the influence of the material parameters, the degree of approximation obtained by using two and three dimensional examples and the monitoring of different quality measures. In this work, the only material parameter that is varied is the Poisson’s ratio (\(\nu \)). Notice that as detailed in [77], the Young’s modulus has no real effect on the resulting highorder meshes because only Dirichlet boundary conditions are considered. Therefore, in all the examples we consider \(E=10^5\), \(E_A = \frac{5E}{2}\) and \(G_A=\frac{E}{2}\) and the Poisson’s ratio is selected within the interval [0.001,0.495].

Incremental Linear Elastic (ILE)

Consistent Incrementally Linearised (CIL)

Nearly Incompressible Mooney–Rivlin (NIMR)

Transversely Isotropic (TI)
The developed code, called PostMesh, has been released as an opensource software under MIT license and is available through the repository https://github.com/romeric/PostMesh.
6.1 Mesh of a mechanical component
It can be observed that the quality of the meshes produced with the ILE isotropic and CIL neoHookean approaches is almost identical, although the CIL neoHookean approach seems to provide better quality for highorder (e.g. \(p=5\)) approximations and for values of the Poisson’s ratio near the incompressible limit. Despite this difference, both approaches are able to produce high quality meshes for any degree of approximation tested. In contrast, the (nonlinear) neoHookean approach fails to produce a highorder mesh for highorder approximations, except for a few cases where a lowquality mesh for \(p=4\) is produced. The quality of the produced meshes for lower order approximations (i.e., \(p=3\),4) is similar to the quality produced by the ILE isotropic and CIL neoHookean approaches, but it is worth noting that the (nonlinear) neoHookean approach also fails in the nearly incompressible region, whereas the ILE isotropic and CIL neoHookean approaches produce the best quality meshes in this scenario.
It should also be noted that, unlike the linearised approaches wherein the internal nodes of the mesh move proportionally to the boundary nodes, in the nonlinear approach the internal nodes can move arbitrarily within the element, and this can in turn affect the quality and approximation property of the produced meshes. In a purely displacementbased formulation, it is not feasible to restrain the movement of internal nodes to a desired proportion. In this context, higher order gradient theories [6, 23, 55] and more elaborate mixed formulations [11, 60], offer a potential future research direction.
Next, we compare the effect of material models presented in Table 2 on the quality of generated meshes, for all the three approaches. Figure 7 shows the quality (minimum scaled Jacobian) as a function of the Poisson’s ratio for all the different models considered in this work, when a polynomial approximation of degree \(p=2\) is employed. For the transversely isotropic model, the negative xaxis is chosen as the direction of anisotropy for the interior elements. For the elements in the boundary, the direction of anisotropy is computed to be perpendicular to the unit normal to boundary edge. This technique is customary in the field of fibrereinforced composites.
The results show that the quality displayed with neoHookean, Mooney–Rivlin and nearly incompressible materials is almost identical for any value of the Poisson’s ratio, whereas a different behaviour is obtained for the transversely isotropic model. The best quality is obtained with the ILE TI model and with a Poisson’s ratio near 0.5. However, it is worth emphasising that a small variability of the quality is obtained in all cases as all simulations provide a highorder mesh with quality belonging to [0.67,0.77].
A different trend is observed, when comparing the results with \(p=3\) to the results with \(p=2\) displayed in Fig. 7. With \(p=3\) the quality of the mesh improves as the Poisson’s ratio is increased, providing the best results always when the incompressible limit is approached. This behaviour is expected in general because when the Poisson’s ratio is taken near 0.5, the imposed displacement on the boundary induces a larger displacement of the interior nodes. In contrast, when a value of the Poisson’s ratio near 0 is considered, the imposed displacement on the boundary induces small displacement on the interior nodes, resulting in more distorted elements (i.e. reduced quality elements). The reason why this expected behaviour was not obtained with \(p=2\) is attributed to the lack of resolution of the displacement field when the coarse mesh considered here, see Fig. 4, is employed with a quadratic approximation. In fact, further simulations not reported here for brevity confirm that with a finer mesh the expected trend is obtained even with a degree of approximation \(p=2\).
In addition, the results show that the quality of the meshes produced with ILE isotropic, CIL and nonlinear approaches is almost identical if a neoHookean model is considered, whereas the use of a transversely isotropic model reveals some differences between the three approaches. The results demonstrate the significance of chosing a welldefined material model like neoHookean (with a quality reported near 0.85), in contrast with a transversely isotropic model (quality reported below 0.6 for any value of the Poisson’s ratio), whose limitations would be discussed shortly. It is worth emphasising that the quality obtained with the Mooney–Rivlin and the NIMR models is almost identical to that produced by the neoHookean model, so that any of the three models is equally suitable to produce high quality meshes in this example, since all these material models are mathematically welldefined.
It can be concluded that the choice of material model does not play a major role, as long as the model is welldefined. As hinted before, unlike the other material models, the transversely isotropic (TI) material defined in Table 2, does not correspond to a polyconvex energy functional, or more specifically, the invariant \(\varvec{N}\cdot \varvec{C}^2\varvec{N} = (\varvec{F}^T\varvec{FN})\cdot (\varvec{F}^T\varvec{FN})\), is not convex with respect to \(\varvec{F}\), and hence under highly large deformations, the model experiences instabilities in the form of loss of ellipticity which can manifest through shearbands, fibre kinking if under compression or fibre debonding if under stretch; cf. [53] and [54], for an intensive study on the loss of ellipticity for this invariant. The latter two phenomena (fibre kinking and debonding) also hold true for the transversely isotropic linear materials. As a consequence, it can be observed that the mean quality of the highorder meshes generated with a transversely isotropic material deteriorates as the order of approximation is increased, compared to the other material models.
Overall, the ILE isotropic approach is found to be the most robust, providing the best or near the best mean quality for all orders of approximation. Also, it is worth noting that for all material models the standard deviation grows as the order of approximation is increased, implying that a good choice of the Poisson’s ratio is more important as the order of approximation is increased.
Again, the results show that the condition number with neoHookean, Mooney–Rivlin and NIMR is almost identical for any value of the Poisson’s ratio, whereas a different behaviour is obtained for the transversely isotropic model. In all cases the condition number increases as the Poisson’s ratio approaches the incompressible limit but it is worth noting that a slightly lower condition number is obtained when the transversely isotropic model is considered, irrespective of the use of ILE isotropic, CIL and nonlinear approaches. This is inherently due to anisotropic nature of the model, as the deformation is not homogenous in every direction and hence the effect of Poisson’s ratio is not equally pronounced for this model. The results with other degrees of approximation are omitted, as exactly the same behaviour is observed.
The last analysis is aimed to compare the computational cost of each formulation with different material models and different orders of approximation p. As it is not feasible to a priori know the number of iterations required by the nonlinear approach to converge, a comparison of the actual computing time is considered here.
Whilst theoretically, the nonlinear approach should cost number of iterations \(\times \) number of increments times more than the linear model, in practice, due to differences in the sparsity pattern and condition number of the system as well as CPU warmup and pipelining, this is not often the case. In fact, comparison of nonlinear against linear approaches is analogous to cold versus hot benchmarking, in that with a higher number of iterations, the processor becomes progressively more accurate with branch prediction and guessing jmp operations, which helps improve processor pipelining. On the other hand, for highly nonlinear problems, with every iteration of the nonlinear analysis the condition number increases, hence impacting the runtime. With this in mind, we report the geometrical mean of 100 runtimes, excluding the timing for the first 10 runs. An inhouse tool similar to Google Benchmark is used for time measurements. For all time measurements, parallelisation has been turned off. Material data and p are deliberately chosen such that the nonlinear analysis would converge. The analysis corresponds to \(\nu =0.4\) with other parameters remaining constant as before.
The computational cost associated with the different material models is clearly related to their tangent operator. In the case of linear elasticity, the tangent operator can be computed at the preprocessing stage. For a neoHookean model the two fourth order identity tensors \(\varvec{I}\otimes \varvec{I}\; (\delta _{ij}\delta _{kl})\) and \(\varvec{\mathcal {I}}\;(\delta _{ik}\delta _{jl}+\delta _{il}\delta _{jk})\), appearing in the tangent operator, are compile time constants. For Mooney–Rivlin, nearly incompressible Mooney–Rivlin and transversely isotropic hyperelastic models, the dyadic products in the tangent operators [see Table 2] are runtime variables and their computation is not always cachefriendly due to unavoidable strided indexing. In fact for complex material models identifying the optimal networks of tensor contraction is not trivial [24, 59]. Using Voigt notation and further permutations, these dyadic products can be transformed to further gemm calls, which eventually may or may not be beneficial. Moreover, the nearly incompressible and transversely isotropic hyperelastic models require computation of cofactors \(\varvec{H}\) and \(\varvec{HH}^T\), at every quadrature point which are all \(O(n^3)\) in computational complexity.
In this example, ILE isotropic approach is found to be the most competitive. This allows to conclude that, for this example, the ILE isotropic approach provides both the best quality and the lowest computational cost compared to other approaches and material models. Furthermore, one should note that the mesh qualities reported here are not indicative of the maximum quality that can be obtained, as the number of load increments is rather kept fixed to facilitate an impartial comparison between different approaches. Finally, it is worth mentioning that although CPU time measurements are always dependent on the implementation, the results reported here provide a qualitative indication of the higher cost associated to a nonlinear approach. The CPU timing, together with the already discussed convergence difficulties of nonlinear approaches for highorder approximations, clearly provides an indication of the limited scope of such an approach for a posteriori highorder mesh generation.
6.2 Mesh around the SD7003 aerofoil
It should be mentioned that, similar to the previous example, it was found that the choice of material model does not have an effect on the quality of the curved meshes and that the transversely isotropic material shows a similar pattern of loss of ellipticity. In the light of these findings, we abandon the comparison of material models for the present example and unless otherwise stated, we only utilise the neoHookean model with its linearised version. In contrast, due to high level of stretching of the meshes considered here, the effect of the number of load increments on the quality of generated meshes will be investigated.
Figure 14 shows the quality of the highorder meshes, measured as the minimum scaled Jacobian, as a function of the Poisson’s ratio and the number of load increments for the ILE isotropic, CIL neoHookean and (nonlinear) neoHookean approaches.
Once more, the nonlinear approach is not able to provide a solution in all cases (i.e. for all values of the Poisson’s ratio and number of load increments). In fact, when it converges, the quality of the nonlinear approach is generally lower than the quality of the ILE isotropic and the CIL neoHookean approaches. It can also be observed that the quality of the meshes produced with the ILE isotropic and the CIL neoHookean approaches is almost identical, for any value of the Poisson’s ratio and for any number of load increments. Finally, the results show that the best quality is obtained for value of the Poisson’s ratio near the incompressible limit and ten load increments approximately. A further increase of the number of load increments does not improve substantially the quality of the meshes but it enables to obtain high quality meshes for slightly lower values of the Poisson’s ratio.
Figure 15 shows the same analysis for meshes with significantly higher level of stretching, namely 100 and 800, for the same degree of approximation, \(p=2\).
For these meshes, the nonlinear approach is only able to provide a result in a few cases. In fact, further numerical experiments show that the higher the stretching, the more cases would display no convergence of the nonlinear approach. In addition, the quality of the meshes produced with the ILE isotropic and the CIL neoHookean approaches is, again, almost identical, for any value of the Poisson’s ratio and for any number of load increments, showing that the conclusions presented do not strongly depend on the level of stretching within the boundary layer.
For the incremental linear elastic and the consistent incrementally linearised approaches, the number of load increments generally improves the mesh quality but the same is not true for the nonlinear approach. If phenomena such as buckling, snapback and snap through are not expected, the nonlinear approach provides the same mesh quality irrespective of the number of increments. However, in the presence of buckling, it is possible to jump through snapback/snapthrough region with fewer load increments, but as the number of load increments is increased the buckling (i.e. snapback/snapthrough regions) cannot be avoided, which in the absence of an arclength technique leads to nonconvergence of the Newton–Raphson method. Furthermore, it is possible for the Newton–Rapshon scheme to converge just prior to the onset of buckling, at the cost of losing quadratic rate of convergence due to illconditioning of the system which essentially emanates from nearly zero Jacobian(s).
Next, the same analysis is performed for higher orders of approximation. Figures 16 and 17 show the quality of the highorder meshes as a function of the Poisson’s ratio and the number of load increments for the ILE isotropic, CIL neoHookean and (nonlinear) neoHookean approaches, for a degree of approximation \(p=4\) and \(p=6\) respectively.
For \(p=4\) the nonlinear approach is unable to converge in the majority of cases. Only for a relatively low stretching, such as 50, and using one load increment, this approach provides a solution for any value of the Poisson’s. ratio. When the stretching is increased to 400, this approach fails to converge even when one increment is used if the Poisson’s ratio is selected near the incompressible limit. For the ILE isotropic and CIL neoHookean approaches the quality of the produced meshes is, once more, almost identical for any value of the Poisson’s ratio, number of increments and stretching. It worth noting that for this order of approximation, the increase in stretching translates into a significant decrease in the maximum quality that can be obtained with the ILE isotropic and CIL neoHookean approaches.
To summarise, Fig. 18 shows the ratio of the scaled Jacobian with 50 load increments over the classical linear elasticity (i.e. single increment), in a logarithmic scale. Note that, due to the logarithmic nature of this measure, a factor of zero implies no improvement and, furthermore, a slight improvement in terms of this factor can imply a significant change in terms of percentage value. For instance, for \(p=6\), and stretching of 1600, the scaled Jacobian improves from 0.0011 for a single increment to 0.3382 for 50 increments. It can be observed, that at high p, it is crucial to increase the number of load increments to obtain good quality meshes, specially if the stretching is also high. In contrast, for loworder approximations the gain obtained by increasing the number of load increments is marginal.
A marginal difference is observed between the ILE isotropic and CIL neoHookean approaches both at lower values of Poisson’s ratio as well as for values near the incompressible limit. As discussed earlier, this figure shows that the nonlinear model is only able to converge when a few load increments are considered. However, note that due to the presence of the geometric stiffness term in the CIL approach, the interior elements are stiffened against heavy distortion and hence the CIL approach typically produces meshes with a slightly better distribution of the quality over the computational mesh, irrespective of the minimum value for quality measures. The results also show the improvement induced by an increase of the Poisson’s ratio. For instance, Fig. 19 (a) shows that the mesh contains a significant number of elements of quality 0.45 when the Poisson’s ratio is 0.11 whereas the minimum quality of the mesh associated to Fig. 19 (c), with a Poisson’s ratio of 0.44, the minimum quality is near 0.75.
Next, the quality of the generated meshes in terms of different measures is studied, namely the measures defined in Equation (27) that are defined in terms of the invariants in Equation (17). The two anisotropic mesh quality measures \(Q_4\) and \(Q_5\) are dropped from the comparison because they are only valid for the transversely isotropic material model, which has been shown to produce low quality meshes in the examples considered. It is worth emphasising that the qualities \(Q_1\) and \(Q_2\) are the same for twodimensional plane strain problems.
Figure 20 shows the quality \(Q_1\) as a function of the Poisson’s ratio and the number of load increments for the mesh with a stretching factor 100 and \(p=6\).
Figure 21 shows the three quality measures \(Q_1\), \(Q_2\) and \(Q_3\) as a function of the Poisson’s ratio for the mesh with \(p=2\), a stretching factor of 25 and using five load increments. The results confirm, numerically, that the quality measures \(Q_1\) and \(Q_2\) are the same for twodimensional plane strain problems. It can also be observed that the ILE and CIL approaches produce meshes of the same quality, irrespectively of the measure used. In addition, the results illustrate that the quality measure \(Q_1\) (and \(Q_2\)) is less influenced by changes on the Poisson’s ratio, compared to \(Q_3\). Finally, the results confirm, once more, the lower quality obtained with the nonlinear approach compared to the ILE and CIL approaches, irrespective of the measure used.
The results show that the quality measure \(Q_1\) (and \(Q_2\)) are less influenced by an increase in the stretching factor, compared to the minimum scaled Jacobian \(Q_3\). In all cases, and for all values of the Poisson’s ratio, the value of \(Q_1\) (and \(Q_2\)) is approximately 0.9, whereas the quality \(Q_3\) can vary from 0.4 to 0.9 depending on the value of the Poisson’s ratio and the level of stretching. When the quality \(Q_3\) is considered, the optimal value of the Poisson’s ratio is clearly dependent on the level of stretching. For low to moderate stretching factors, a Poisson’s ratio near the incompressible limit provides the highest quality whereas for very high stretching factors it is better to consider values in between 0.3 and 0.4.
This example, shows a different behaviour of the ILE isotropic and CIL neoHookean approaches. The CIL approach shows a significant deterioration of the quality measure \(Q_1\) (and \(Q_2\)) near the incompressible limit, whereas the ILE isotropic approach maintains a high quality for all values of the Poisson’s ratio.
Compared to the previous problem, there is a significant increase in the degrees of freedom and hence the overhead of function calls is insignificant compared to the actual cost of computation.
For highly stretched meshes, the Newton–Raphson scheme loses quadratic convergence. The increased number of iterations required and the higher cost of each iteration, due to illconditioning, makes the cost of the nonlinear approach significantly higher. The ILE (isotropic and TI) approaches are found to be the most competitive. This allows to conclude that, as in the previous example, the ILE approaches provide both the best quality and the lowest computational cost compared to other approacher and material models.
The last study for this example, involves a pconvergence analysis in order to illustrate the optimal approximation properties of the produced meshes. Given a smooth function defined in Cartesian coordinates, the strategy consists on computing the exact value of the solution at the mesh nodes. Then, the error between the approximated solution, interpolated from the nodal values, and the exact solution is computed at each integration point to compute the error in the \(\mathcal {L}^2(\varOmega )\) norm.
The results show the expected exponential convergence in the approximation of a smooth function. In addition, it is interesting to observe that the error is almost identical for the ILE and CIL approaches. This conclusion is in line with the previous analysis where it was shown that the quality of the meshes produced with the ILE and CIL approaches is almost identical, except in some extreme cases considering highly stretched meshes, highorders of approximation and values of the Poisson’s ratio near the incompressible limit. In contrast, the CIL TI approach, which was shown to produce lower quality for highorder approximation shows a deterioration in the convergence rate, which illustrates the importance of producing high quality meshes for finite element analysis. Finally, the results also show the ability to preserve the approximation properties independently on the level of stretching.
6.3 Mesh around the NASA almond
The next example considers a tetrahedral mesh around the NASA almond, a popular geometry for benchmarking 3D radar cross section computations in computational electromagnetics [22, 76]. Figure 26 shows the linear surface mesh of the almond, the highorder surface mesh corresponding to a degree of approximation \(p=4\) and a cut of the highorder volume mesh. The linear mesh contains 6247 elements, 1288 nodes and 688 faces on the almond. The corresponding highorder mesh with \(p=6\) contains 233,205 nodes and 16,420 nodes to be projected over the true almond geometry to obtain the Dirichlet boundary conditions of the solid mechanics problem.
Compared to the twodimensional results of the isotropic meshes in Sect. 6.1, similar conclusions are derived here. First, the quality of both the meshes produced with the ILE isotropic and CIL neoHookean approaches is similar, although the ILE isotropic provides better quality near the incompressible limit and, for some particular choices of the approximation degree, for the whole range of values of the Poisson’s ratio (e.g., for \(p=5\)). As shown in previous examples, the nonlinear approach produces good quality meshes for loworder approximations (i.e., \(p=2\),3). For \(p=4\) a valid mesh is only obtained for values of the Poisson’s ratio between 0.1 and 0.4, and no convergence is obtained if the order of approximation is further increased.
Although the actual value of the quality is different, depending on the selected measure, the qualitative behaviour is the same compared to the quality \(Q_1\). As reported earlier with the two dimensional examples, the quality measure that produces a lower absolute value is the scaled Jacobian, \(Q_3\), traditionally used by the highorder mesh generation community. This is attributed to the motion resulting from an imposed boundary displacement that results from projecting the highorder nodes to the true CAD surface. In this scenario, the volumetric deformation related to \(Q_3\), is much more important than the deformations related to \(Q_1\) and \(Q_2\).
The results confirm that, contrary to two dimensional plane strain problems, the quality measures \(Q_1\) and \(Q_2\) are different. It can be observed that the ILE and CIL approaches produce meshes of similar quality, irrespective of the measure considered. In addition, the results illustrate that the quality measure \(Q_1\) is less influenced by the Poisson’s ratio whereas the quality \(Q_3\) shows a major dependence on this material parameter. Finally, the results shows that for loworder approximations the nonlinear approach can produce meshes of slightly better scaled Jacobian compared to the ILE and CIL approaches although when the quality measures \(Q_1\) and \(Q_2\) are used, the nonlinear approach produce the lowest quality meshes compared to the ILE and CIL approaches. This is again due to the nonproportional movement of the nodes in the nonlinear approach, which results in distortion of edges and faces of the element, despite a reasonable volumetric deformation being maintained.
If a higher order of approximation is considered, say \(p=5\), the nonlinear approach fails to converge for any value of the Poisson’s ratio, as illustrated in Fig. 27. A comparison of the different quality measures for the ILE and CIL approaches is shown in Fig. 31.
The results reveal important differences between the ILE and CIL approaches and illustrate the robustness of the ILE approach as the quality is significantly less dependent on the value of the Poisson’s ratio selected, compared to the CIL approach. In fact, the results show that high quality meshes can be obtained for the ILE approach with any value of the Poisson’s ratio, even with a value near 0, whereas a substantial decrease in the quality is observed if a Poisson’s ratio near 0 is selected for the CIL approach.
As done in previous examples, the CPU time has been normalised with respect to that of classical linear elasticity and the geometrical mean of 100 runtimes, excluding the timing for the first 10 runs, is reported. Compared to previous twodimensional examples, the number of degrees of freedom is now significantly larger for a single core and, therefore, the cost of actual computation dominates over secondary effects such as inlining and branch prediction. The systems of linear equations are now solved using the Multifrontal Massively Parallel Solver (MUMPS). It is interesting to observe that, despite these differences compared to the twodimensional examples, similar conclusions are obtained from the CPU time analysis. Once more, both the ILE approaches are found to be the most competitive and the nonlinear approaches the most computationally expensive. These results, together with the quality study presented in this section, enables to conclude that the ILE and CIL approaches are recommended for producing highorder curvilinear meshes from an initial linear mesh.
The results show the expected exponential convergence in the approximation of a smooth function. In addition, it is interesting to observe that the error is almost identical for the ILE and CIL approaches. Once more, the CIL TI approach shows a slight deterioration in the rate of convergence for highorder approximations due to the lower quality of the meshes produced with this approach. This result in fact pinpoints the importance of choosing a welldefined polyconvex material model, in the context of a posteriori mesh generation.
6.4 Meshes around full aircraft configurations
The next examples consider meshes around two full aircraft configurations, showing the capability of the proposed unified approach for generating meshes around realistic geometries of interest to the computational electromagnetics and computational fluid dynamics communities.
First, a tetrahedral mesh around a generic Falcon aircraft is considered. The linear mesh has 185,191 elements, 35,875 vertices and 16,922 triangular faces on the aircraft to be projected on the true CAD geometry to obtain the Dirichlet boundary condition for the solid mechanics problem. The corresponding CAD geometry has 54 surfaces with 240 intersection curves. For an interpolation degree of \(p=3\), there are 876,988 nodes in the domain and 76,151 nodes on the aircraft that require projection.
Figure 34 shows the linear surface mesh of the aircraft, the highorder surface mesh corresponding to a degree of approximation \(p=3\) and a cut of the highorder volume mesh. The problem is solved using the CIL Mooney–Rivlin approach with \(\nu =0.45\) and 20 load increments. The minimum Scaled Jacobian for this mesh is \(Q_3=0.337\) and there are 181,251 elements (i.e. 97.87 percent of the total number of elements) for which \(Q_3>0.9\). The minimum values of the other two quality measures, accounting for fibre and surface deformations, are \(Q_1 = 0.605\) and \(Q_2 = 0.467\).
The problem is solved using the ILE isotropic approach with \(\nu =0.45\) and 100 load increments. The minimum values of the three quality measures for this mesh are \(Q_1=0.482\), \(Q_2=0.377\) and \(Q_3=0.329\). Moreover there are only 11 elements with a quality \(Q_3<0.9\).
Finally, a boundary layer tetrahedral mesh around the DLRF6 transport configuration with a stretching of 317 is considered. The boundary layer has been constructed such that the final mesh is suitable for a compressible NavierStokes simulation up to a Reynolds number of approximately \(Re=4\times 10^7\). The linear mesh has 4,482,662 elements, 787,712 vertices and 110,458 triangular faces on the aircraft. Two curved boundary layer meshes are generated for this geometry with \(p=3\) and \(p=4\), respectively. The resulting highorder mesh with \(p=3\) has 20,434,689 nodes with 498,590 nodes on the aircraft and the \(p=4\) mesh has 48,279,087 nodes with 885,712 nodes on the aircraft. Both meshes are produced using the ILE isotropic approach. The number of increments are chosen such that a balance is kept between computational cost and final quality of the computational mesh. This corresponds to 60 and 30 increments with a minimum scaled Jacobian of 0.06 and 0.02 for \(p=3\) and \(p=4\), respectively. However, for both meshes, 99.5 \(\%\) of the elements have scaled Jacobian above 0.8.
6.5 Meshes of complex mechanical components
Two complex threedimensional mechanical components are considered in this section. The Poisson’s ratio for all the examples considered in this section is chosen as \(\nu =0.45\).
7 Conclusions
A unified approach for the generation of highorder curvilinear meshes derived via a solid mechanics analogy has been presented. This proposed theoretical and computational approach encompasses the incremental linear elastic approach (wherein only the geometry is updated incrementally) and the fully nonlinear approach, both previously applied in the context of a posteriori highorder mesh generation. In addition, the new incrementally linearised elasticity formulation (wherein the geometry, the tangent operator and the stresses are updated incrementally), not previously applied to generate curvilinear highorder meshes, is included within this unified approach. The material parameters are calibrated such that the tangent operators of all the aforementioned approaches with various material models are identical in the reference configuration, i.e. for the (undeformed) mesh with planar faces or edges. The derivation of all the approaches, based on energy principles, is used to propose mesh quality measures based on independent invariants of the strain energy density. The relation of the proposed quality measures with indicators previously used in the context of highorder curved mesh generation is discussed.
Several numerical examples are presented in both two and three dimensions, including realistic geometries of interest to the solids, fluids and electromagnetics communities. A detailed comparison of all the methodologies is made, including the quality of the generated highorder meshes, the influence of material parameters and load increments on the resulting meshes, the computational cost and the approximation properties of the meshes when applied to an isoparametric finite element formulation.
In terms of the material parameters, the use of a Poisson’s ratio near the incompressible limit is generally advised in order to maximise the quality of the resulting mesh. For isotropic meshes, a low number of increments (e.g. five increments) is typically sufficient to obtain the maximum possible quality, whereas for highly stretched meshes and for highorders of approximation (i.e. \(p>4\)) a higher number (e.g. 40 increments) is needed to obtain good quality meshes. Both factors are in fact related as the results show that a higher number of increments is needed when the Poisson’s ratio approaches the incompressible limit.
When the material parameters are kept the same, all the linearised approaches, in particular, the incremental linear elastic and the consistent incrementally linearised approach produce meshes of very similar quality and only small differences are observed for highly stretched meshes when highorders of approximation are used and the Poisson’s ratio approaches the incompressible limit. In contrast, the nonlinear approach has been found to produce poor quality elements when a highorder approximation is utilised. The nonproportional displacement of interior nodes with respect to the imposed displacement of boundary nodes has a significant negative impact on the convergence of the nonlinear solver. Only for loworder approximations has the nonlinear approach shown robustness and the ability to produce good quality meshes. The importance of having a welldefined internal energy for the nonlinear material model has been illustrated using the transversely isotropic hyperelastic material. For highly stretched meshes, buckling can be expected in the nonlinear analysis and the Dirichletdriven nature of the problems demands a sophisticated and expensive arclength technique to guarantee convergence, hindering its practical use in an a posteriori mesh generation framework.
The three quality measures proposed for isotropic materials, namely, \(Q_1\) related to fibre maps, \(Q_2\) related to surface maps and \(Q_3\) related to volume maps, show a similar trend with respect to the material parameters. In fact, the first two quality measures are identical for two dimensional plane strain problems. For all the examples considered, \(Q_3\) is the most impactful indicator, which corresponds to the socalled scaled Jacobian traditionally used by the highorder mesh generation community.
In terms of the computational cost, the nonlinear approach is much more expensive than the linearised approaches. For highly stretched meshes, where the Newton–Raphson scheme may lose its quadratic convergence due to illconditioning of the system, a higher number of iterations is required and the solver time is drastically increased. The linearised approaches are not only much more economical but, in addition, more robust and produce better quality meshes.
Finally, the approximation properties of the resulting meshes have been assessed and the results show that a similar quality of mesh (as indicated by \(Q_1\), \(Q_2\) and \(Q_3\)) translates in similar interpolation errors (i.e. the quality indicators have been shown to be well chosen).
Footnotes
 1.The cross product ( Open image in new window ) between two tensors is defined aswhere \(\varvec{\xi }\) is the third order permutation tensor, see [11, 12] for a list of properties.
 2.
 3.
Note that for plane strain problems, the first two isotropic invariants are identical i.e. \(I_1=I_2\).
 4.To satisfy polyconvexity in a transversely isotropic material, the independent invariant \(I_5\) is instead given by$$\begin{aligned} I_5 = \varvec{HN}\cdot \varvec{HN} = (\text {det}\varvec{C}) ( \varvec{N}\cdot \varvec{C}^{1}\varvec{N}). \end{aligned}$$
Notes
Acknowledgments
The first author gratefully acknowledges the financial support received through The Erasmus Mundus SEED program. The second author gratefully acknowledges the financial support provided by the Sêr Cymru National Research Network in Advanced Engineering and Materials. The third author gratefully acknowledges the financial support received through The Leverhulme Prize awarded by The Leverhulme Trust, UK.
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