HDGNEFEM with Degree Adaptivity for Stokes Flows
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Abstract
The NURBSenhanced finite element method (NEFEM) combined with a hybridisable discontinuous Galerkin (HDG) approach is presented for the first time. The proposed technique completely eliminates the uncertainty induced by a polynomial approximation of curved boundaries that is common within an isoparametric approach and, compared to other DG methods, provides a significant reduction in number of degrees of freedom. In addition, by exploiting the ability of HDG to compute a postprocessed solution and by using a local a priori error estimate valid for elliptic problems, an inexpensive, reliable and computable error estimator is devised. The proposed methodology is used to solve Stokes flow problems using automatic degree adaptation. Particular attention is paid to the importance of an accurate boundary representation when changing the degree of approximation in curved elements. Several strategies are compared and the superiority and reliability of HDGNEFEM with degree adaptation is illustrated.
Keywords
Hybridisable discontinuous Galerkin NURBSenhanced finite element method Degree adaptivity StokesMathematics Subject Classification
65N08 65N12 65N22 65N301 Introduction
Early work on mesh and degree adaptivity schemes for the finite element method [30, 44, 65] already showed the advantages of adaptive schemes to achieve a required accuracy in an economic manner. The use of mesh adaptive methods is substantially more extended due to the popularity of loworder methods in the computational mechanics community. This is largely due, as discussed later, to the fact that mesh adaptation is easier to implement, compared to degree adaptivity, in standard finite element codes. But, with recent needs on high fidelity simulations for fluids and wave propagation phenomena [14, 25, 63], the interest in degree adaptive (or the combination of mesh and degree adaptivity) processes has increased [4, 23, 24, 31].
One of the main reasons for the increasing popularity of degree adaptive schemes in the last years is the rise of discontinuous Galerkin (DG) methods as a viable alternative for convection dominated flow and wave propagation problems [13, 15, 21, 26, 43, 58]. In a standard continuous Galerkin framework, the implementation of variable degree of approximation is cumbersome, whereas its application in a DG context is straightforward due to the weak imposition of the continuity of the solution by means of numerical fluxes. Despite traditional DG methods have not been able to consistently prove its superiority against loworder techniques traditionally employed in industry (e.g. finite volume methods), the recently proposed hybridisable DG (HDG) [11] has shown its superiority compared to traditional DG methods [9, 27, 33]. The ability to substantially reduce the number of degrees of freedom combined with the possibility to obtain a postprocessed solution that converges at a faster rate to the exact solution are the two main properties of HDG methods behind its superiority compared to other DG methods [10, 12, 38, 56]. Moreover, this is achieved while preserving the wellknown advantages of DG for stabilising convection and circumventing the socalled LadyzhenskayaBabuškaBrezzi (LBB) condition in the incompressible limit.
A key aspect in any adaptive scheme is the ability to devise cheap and reliable error measures for a given numerical solution in order to decide the regions where a more accurate solution is required [1]. Error indicators and error estimators are typically employed to asses the error of a simulation with an adaptive framework [29]. Error indicators are computationally inexpensive but they are problem dependent whereas error estimators are considerably more expensive but more general [19, 46, 47]. A cheap, general and reliable error estimator was proposed in [23, 24] by exploiting the ability of the HDG method to construct a postprocessed solution, more accurate than the HDG solution.
One of the aspects that is normally ignored when devising degree adaptive schemes is the geometric representation of domains with curved boundaries. Despite it is now well known that a poor representation of the geometry can have an important effect on the results of a finite element simulation [2, 7, 54, 60], the most extended practice consists on maintaining the shape of the elements during the degree adaptive process [23, 24, 31]. In the majority of cases, a polynomial representation of the boundary is selected whereas the polynomial degree of the functional approximation changes at each iteration of the degree adaptive scheme.
This work analyses and discusses three approaches to perform a degree adaptive process in domains with curved boundaries. The first one corresponds to the approach typically employed in practice, consisting of fixing the shape of the curved elements and changing the degree of the functional approximation as dictated by the degree adaptivity procedure. The second approach proposed in this work is to employ the socalled NURBSenhanced finite element method (NEFEM) that enables to exactly represent the geometry of the computational domain, given by a CAD model, irrespectively of the degree of the polynomials used to approximate the solution. The third approach, despite not considered useful from a practical point of view, consists of changing the geometry representation of the computational domain to represent with the same degree of polynomials both the geometry and the solution at each iteration of the degree adaptive process. This approach is not considered of interest from a practical point of view because it requires communication with the CAD model at each iteration and regeneration of nodal distributions for curved elements.
The second approach proposed here considers, for the first time, the combination of the socalled NURBSenhanced finite element method (NEFEM) and the HDG rationale. The resulting method combines all the advantages of both methods, that is the efficiency of HDG and the ability of NEFEM to decouple the functional approximation from the geometric representation, usually tied in traditional isoparametric implementations.
A number of numerical examples is considered in order to compare the different degree adaptivity approaches. Furthermore, this work presents a simple idea to verify computational methods that are able to use different degrees of approximation for the solution in different elements. The idea is based on an existing local a priori error estimator developed in [18] for elliptic problems.
The remainder of the paper is organised as follows. Section 2 briefly presents the model problem considered (i.e. Stokes flows) and the HDG formulation. The spatial discretisation of the HDG weak formulation is presented in Sect. 3 for both isoparametric and NEFEM, with particular emphasis on the differences between both formulations. The details about the proposed error estimator and degree adaptivity process proposed are presented in Sect. 4, including a discussion of the three approaches considered to perform a degree adaptive process. In Sect. 5 a simple technique to verify the implementation of a solver with variable degree of approximation is presented and used to test the implementation of the HDG code for Stokes flows with isoparametric and NEFEM. Section 6 presents a comparison of the different degree adaptive approaches and a number of numerical examples are used in section to show the potential of the proposed approach. Finally, Sect. 8 summarises the main conclusions of the work that has been presented.
2 Hybridisable Discontinuous Galerkin for Stokes Flow
2.1 Problem statement
2.2 HDG Weak Formulation
3 Spatial Discretisation
This section presents the discretisation of the HDG weak forms derived in the previous section. Both the standard isoparametric and the socalled NEFEM formulations are presented. Special attention is paid to the differences between both formulations as this represents the first time NEFEM is considered in an HDG framework.
3.1 Isoparametric Elements
It is worth noting that in general, when the physical element \(\varOmega _e\) is curved, the isoparametric mapping is nonlinear and the approximation defined in the reference element do not induce a polynomial interpolation in the physical space. In addition, the computational element \(\varOmega _e^h\) is just an approximation of \(\varOmega _e\), see [53] for a detailed discussion.
3.2 NEFEM Elements
In NEFEM, the boundary of the computational domain \(\partial \varOmega \) is exactly represented by NURBS. In what follows, in order to simplify the presentation and without loss of generality, the NURBS are restricted to two dimensional problems, see [52] for a detailed description of the three dimensional case. An edge is given by \(\varGamma _e := \varvec{C}([\lambda ^e_a,\lambda ^e_b])\), where \(\varvec{C}\) is the NURBS boundary parametrisation and \(\lambda _a\) and \(\lambda _b\) are the parametric coordinates (in the parametric space of the NURBS) of the end points of \(\varGamma _e\).

The exact description of the computational domain is considered by means of its NURBS boundary representation.

The approximation of the elemental variables directly in the physical space, with Cartesian coordinates.

The approximation of the trace of the velocity is defined in the parametric space of the NURBS. It is worth noting that other options could be considered such as defining the approximation directly in the physical space. The main advantage of defining the approximation in the parametric space of the NURBS is that the number of unknowns remains the same as in the isoparametric formulation. In contrast, if the approximation of this variable is selected in the physical space it would require further degrees of freedom [53, 54].
Using the mapping in Eq. (26) and the NURBS boundary representation given by \(\varvec{C}\), the integrals appearing in the weak form of the local problems are transformed to the reference rectangle and the parametric space of the NURBS respectively. Then, the nodal interpolations given by Eqs. (22), (23), (24) and (25) are introduced, leading to a system of equations similar to Eq. (20). Analogously, the global problem with NEFEM leads to a global system of equations similar to Eq. (21).
4 Error Estimation and Adaptivity
In HDG, the possibility to obtain a postprocessed solution [11] that converges at a higher rate (i.e. \(k+2\)) than the HDG solution, not only provides a higher accurate solution to the problem at hand but it can also be used to build an inexpensive, reliable and computable error estimator [23, 24]. In this section, particular attention is paid to the fact that, when the degree of approximation is changed in a curved element, a choice must be made regarding the geometric definition of the element.
The adaptive procedure consist on solving the Stokes problem using the HDG formulation as described in Sect. 3 and estimating the required degree of approximation in each element according to Eq. (29). The process is repeated until convergence is achieved, meaning that the error in each element \(\varepsilon _e\) is lower than the desired error \(\epsilon \).
4.1 Geometry Update
The technique described to drive a degree adaptive process only focuses on the degree of approximation used for the functional approximation, but in the presence of curved boundaries it is known that highorder approximations of both the solution and the geometry are required to exploit the full potential of a highorder method [2, 17, 34, 55]. This aspect is usually ignored as degree adaptive procedures are applied to problems involving polygonal boundaries, see for instance [16, 22, 45, 61]. Here three options are discussed and assessed and compared later using numerical examples.
Remark 1
It is important to note that the first strategy, where the geometry remains unchanged, does not guarantee the convergence of the numerical solution to the physical solution in domains with curved boundaries because the distance between the computational domain and the physical domain does not converge to zero with as the degree of approximation is increased, see [6, 49] for more details. For the second strategy, proposed here, convergence to the physical solution is guaranteed because no geometrical error is introduced [54]. Finally, for the third approach, convergence is also guaranteed if the distance between the computational boundary and the physical boundary tends to zero as the order of the approximation is increased and the derivatives of the isoparametric mapping up to order \(k+1\) are bounded by \(h^s\), for \(s=2,\ldots ,k+1\) [6, 49], where h denotes the characteristic element size. It is worth noting that a specifically designed nodal distribution for curved elements is required in the third approach to guarantee that the second hypothesis is fulfilled [6].
5 Validation of the HDG Formulation with Variable Degree of Approximation
The first example provides a novel and simple technique to fully validate a solver that employs variable different degree of approximation in different elements for the solution of elliptic problems. The idea consists of utilising the local a priori error estimate of Eq. (28) that states how the error, measured in an element, decreases when the mesh is refined.
Again, the optimal rate of convergence is obtained in all cases for both \(\varvec{L}\) and p (rate \(k+2\)).
6 Comparison of Degree Adaptivity Strategies
The same model problem employed in the previous example is utilised to compare the strategies described in Sect. 4.1 to update the geometry during a degree adaptive process. The computational domain selected, shown in Fig. 7, features an oscillatory boundary and represents a common problem encountered in biological transport applications, see for instance [50]. More precisely, the curved part of the boundary is given by the curve \(f(x) = (1 + \cos (5\pi x))/10\).
6.1 No Geometric Update
As a linear approximation of the geometry is well known to be not suitable when high order functional approximations are considered [2, 7, 54, 60], the same experiment is repeated by using a more accurate boundary representation. The plots in Fig. 10b–d show the evolution of the maximum estimated error in each element and the maximum exact in each element for quadratic, cubic and quartic approximation of the geometry. In all cases it is clearly observed that the error estimator is not reliable because the adaptive process converges but the exact error is more than one order of magnitude higher than the desired error.
6.2 NEFEM HDG
The strategy proposed in this work consists of utilising NEFEM, where the geometry is always given by its CAD boundary representation, irrespective of the degree of the functional approximation. In the context of a degree adaptive process, this means that no communication the CAD model is required as the exact boundary representation is already used by the NEFEM solver.
The process starts with a degree of approximation \(k=1\) in all elements. At each iteration the degree of the functional approximation is adapted according to the strategy presented in Sect. 4 and a new nodal distribution is generated for each curved element.
Remark 2
As discussed in Sect. 4.1, an alternative, not employed in practice due to the high cost induced by the regeneration of the mesh at each iteration of the adaptive process, consists of changing both the degree of approximation for the solution and for the geometry during the adaptivity process, as illustrated in Fig. 3.
In addition, it is worth emphasising that the isoparametric approach requires communication with the CAD model in each iteration to regenerated the highorder nodal distribution. These nodal distributions in curved elements must be specifically designed to ensure optimal convergence of the isoparametric approach [3, 5], while for NEFEM the Cartesian approximation of the solution ensures that the accuracy of the approximation is much less sensitive to the quality of the nodal distribution.
7 Numerical Examples
This section presents four numerical examples to illustrate the potential of NEFEM when combined with HDG to perform a degree adaptive process. The examples involve geometries with curved boundaries and where coarse meshes are considered to show the robustness of the proposed methodology. In all the examples the highorder isoparametric and NEFEM meshes are generated using the techniques described in [48, 64] and [57] respectively.
7.1 Flow in a Channel with Randomly Distributed Ellipses
A coarse mesh with 2,443 triangular elements is first considered. As no analytical solution is available, a reference solution is computed in a much finer mesh with 28,150 elements and by employing a degree of approximation \(k=4\). This reference solution is used to measure the accuracy of the adaptive computations performed in much coarse meshes.
When the same computation is performed by considering a cubic approximation of the geometry (not reported for brevity), the adaptive process converges again in five iterations. he highest order of approximation used in a few elements is now \(k=7\), indicating that a different geometric representation leads to a different degree of approximation required to achieve convergence. In addition, the error estimator is again not reliable as there are 15 elements where the reference error is above the desired error.
To show the potential of NEFEM in this scenario, an adaptive process is performed employing the coarse mesh with 2,443 triangular elements and starting with a degree of approximation \(k=1\). The adaptive process converges in four iterations. The final degree of approximation used in each element is shown in Fig. 18a, with two elements having the maximum degree of approximation, \(k=6\), required to achieve the desired error. The estimated and reference errors, depicted in Fig. 18b, c, respectively, shows a consistent behaviour that illustrates the reliability of the proposed strategy to estimate the error due to the use of the exact boundary representation. It is worth noting that in the majority of the elements surrounding the ellipses a degree of approximation \(k=3\) is enough to obtain the required error, illustrating why cubic isoparametric elements outperformed the use of quadratic elements in the previous computations.
The velocity field computed with NEFEM on the mesh shown in Fig. 18a is depicted in Fig. 19.
The results show that a cubic approximation of the geometry is not enough because the difference between the estimated and exact error in the final computation with cubic elements is almost an order of magnitude. This suggests, once more that the initial mesh has to be preadapted to achieve a reliable degree adaptive process.
Finally, Fig. 22 shows the results obtained with NEFEM using the coarse mesh with 2,443 elements and starting with a linear approximation of the solution \(k=1\). The robustness of the proposed approach is clearly illustrated as convergence of both the estimated and exact errors is achieved in the coarse mesh even for a desired error of \(0.5 \times 10^{4}\). It is worth emphasising that with NEFEM the adaptive process provides a reliable error estimator even when the desired error is several orders of magnitude lower than the error of the computation in the first mesh. The results clearly indicate that no preadaptation of the mesh is required with NEFEM as the geometry is exactly represented irrespectively of the spatial discretisation. Therefore, the adaptive process is purely driven by the functional approximation and not by the geometric error as it happens with an isoparametric formulation.
7.2 Flow in a Channel with Wavy Boundaries
7.3 Flow in a Porous Media
It is worth noting that a linear degree of approximation is used in many elements in contact with curved boundaries. This shows that the proposed adaptivity strategy is completely driven by the complexity of the solution and not by the complexity of the geometry.
7.4 Flow in a Channel with Thin Obstacles
The flow in a channel with a number of thin obstacles is considered. The thickness of the obstacles is approximately 0.08 whereas the minimum element size of the mesh that has been generated, using the technique proposed in [57], is 0.32. The degree adaptive process is started, as in previous examples, with a linear approximation of the solution and convergence is achieved in four iterations. The final degree of approximation in each element is represented in Fig. 26a. Figure 26 shows a detailed view of a region in the channel showing the elements near the end of some of the obstacles. This plot shows not only that the element size is independent on the geometric complexity but it also demonstrates the robustness of the proposed degree adaptive technique. The adaptivity process is clearly driven only by the complexity of the solution as a different degree of approximation is employed in elements with almost identical geometric complexity due to the different complexity of the solution.
8 Concluding Remarks
A new degree adaptive methodology that combines the advantages of the HDG formulation and NEFEM has been presented. The proposed method results in a cheap and reliable error estimator due to the cheap computation of a postprocessed solution provided by HDG and the ability to exactly represent curved boundaries irrespectively of the polynomial degree used for the functional approximation that is characteristic of NEFEM.
The proposed approach is compared against two alternative options to perform degree adaptivity. The first approach, broadly used in practice, consists of keeping the shape of the curved elements during the degree adaptivity process. It is found that this approach leads to an unreliable error estimator. The numerical examples show that even when the estimated error is below the required tolerance, the exact error can be orders of magnitude higher. The second approach, not used in practice, consists of changing the shape of the curved elements during the adaptive process. The main drawback is its high cost due to the need to constantly communicate with the CAD model and regenerate the nodal distributions for curved elements.
The proposed approach considers, for the first time, the implementation of the NEFEM rationale in an HDG framework. A number of numerical examples have been presented to compare the performance of the proposed methodology and to shows its superiority on a number of problems involving domains with curved boundaries.
Notes
Acknowledgements
The authors gratefully acknowledge the financial support of the Ministerio de Economía y Competitividad (Grant Number DPI 201451844C22R). The first author also gratefully acknowledges the financial support provided by the Sêr Cymru National Research Network for Advanced Engineering and Materials (Grant Number NRN045). The second author is also grateful for the financial support provided by the Generalitat de Catalunya (Grant Number 2014SGR1471).
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