An unstructured immersed finite element method for nonlinear solid mechanics
 Thomas Rüberg^{1, 2}Email author,
 Fehmi Cirak^{3} and
 José Manuel García Aznar^{1}
https://doi.org/10.1186/s4032301600775
© The Author(s) 2016
Received: 29 March 2016
Accepted: 7 July 2016
Published: 22 July 2016
Abstract
We present an immersed finite element technique for boundaryvalue and interface problems from nonlinear solid mechanics. Its key features are the implicit representation of domain boundaries and interfaces, the use of Nitsche’s method for the incorporation of boundary conditions, accurate numerical integration based on marching tetrahedrons and cutelement stabilisation by means of extrapolation. For discretisation structured and unstructured background meshes with Lagrange basis functions are considered. We show numerically and analytically that the introduced cutelement stabilisation technique provides an effective bound on the size of the Nitsche parameters and, in turn, leads to wellconditioned system matrices. In addition, we introduce a novel approach for representing and analysing geometries with sharp features (edges and corners) using an implicit geometry representation. This allows the computation of typical engineering parts composed of solid primitives without the need of boundaryfitted meshes.
Keywords
Background
Conventional finite element methods (FEM) are an irreplaceable tool for the numerical analysis of a variety of physical and engineering problems. They rely on a conforming mesh which approximately matches the domain boundary and material interfaces. For this reason, mesh generation is an essential part of the workflow in FEMbased analyses [1]. Although the procedure is wellestablished, often the use of a boundaryconforming mesh can be limiting or even prohibitive. Fluidstructure interaction, large elastic deformations and shape optimisation are some applications where mesh entanglement can cause severe difficulties for conventional FEM.
In the last two decades or so, a number of finite elementbased numerical methods have been introduced in order to eliminate the need for boundaryconforming meshes. Here, we restrict ourselves to immersed methods, also known as embedded of fictitious domain methods, that operate with a geometryindependent mesh, in the line of [2–6]. Since the mesh of an immersed domain method does not conform with the boundary of the physical domain, one of these methods’ main difficulties is the application of boundary conditions. Here, we choose Nitsche’s method [7] for the weak enforcement of Dirichlet boundary conditions because it gives optimal convergence rates without incurring major implementation difficulties. Moreover, the use Langrange multipliers together with its numerical intricacies, such as the fulfilment of the LBBcondition [8], are avoided. For alternative approaches, see [4, 9–14] among others.
A major difficulty of nonbodyfitted methods is the accurate integration of the arising volume and surface integrals. Here, we make use of a tessellation concept which allows to incorporate standard, Gauß quadrature schemes. In the course of this development, a technique is presented which enables the representation of sharp domain features by performing constructive solid geometry (CSG) modelling directly on the embedding mesh. This approach poses a clear advantage in comparison to the conventional methods of geometry resolution because these sharp features are accurately reproduced and not chamfered even on coarse meshes.
Another pitfall of immersed finite element methods is the loss of numerical stability in cases where the intersection of a shape function support with the physical domain becomes very small. This issue has been successfully addressed in the context of bspline finite elements [6, 12, 15]. In this work, we build up on this concept of constraining critical degrees of freedom and apply it to Lagrangian basis functions on unstructured meshes. Note that Burman et al. [16] introduced an alternative approach, the socalled ghostpenalty stabilisation method, which is based on an augmented bilinear form. Strongly related to stability are the method’s parameters and we show how to choose these parameters in the context of the introduced stabilisation techniques.
The method we present here is based on our previous works [6, 17–19] and related to [2, 3, 16, 20, 21]. Although, as shown in the cited works, the method can be transferred to many physical applications, we focus on the problem class of nonlinear elasticity.
Weak enforcement of boundary and interface conditions
Boundary value problems of nonlinear solid mechanics
Material interfaces
Immersed finite element method
Finite element discretisation
Above expressions hold analogously for interface problems. The main difference is that the two fields \(\varvec{u}_1\) and \(\varvec{u}_2\) are approximated in fashion of (20) independently on the same background mesh of \(\Omega _\square \) which encompasses both subdomains \(\Omega _1\) and \(\Omega _2\). Consequently, the elements which are traversed by the material interface approximate both fields since the FE shape functions of the entire element are used even though the fields are only defined up to the interface on their respective side of the domain. Using two sets of shape functions on these elements allows us to represent a discontinuous derivative of the FE solution and can thus be compared to the element enrichment of XFEM [11]. A good illustration of this implementation detail can be found in [2].
Signed distance functions
 1
\(\tau _I \cap \Omega = \emptyset \), the element is completely outside of \(\Omega \) and can be ignored,
 2
\(\tau _I \cap \Omega = \tau _I\), the element is completely inside and its treatment is straightforward as in any geometryconforming FEM,
 3
\(\tau _I \cap \Gamma \ne \emptyset \), the element is traversed by the domain’s boundary and requires special consideration.
The representation (23) can be of higher polynomial degree, given by NURBS patches [1, 14, 27] or subdivision surfaces [28, 29]. But the computation of the coefficients \(d_K\) in (25) and the quadrature described below are nontrivial tasks if the \(\sigma _J\) have a degree higher than linear simplex elements (straight lines in two or flat triangles in three dimensions). In that case, the computation of the distances \(d_K\) requires the solution of nonlinear equations, see, for instance, [30]. In the rest of this work, the \(\sigma _J\) are always linear \((n_d1)\)simplex elements. Moreover, once only piecewise linear elements are used for the surface representation (23), the optimal convergence rate of any higherorder method is impeded by this geometry approximation error, see [8].
Once the function \({{\mathrm{dist}}}_\Gamma \) has been determined, the above classification of volume elements \(\tau _I\) is carried out by means of the nodal values \(d_K\) of the distance function: if all \(d_K\) of the element \(\tau _I\) are strictly positive (negative), the element is inside (outside) of the domain. If a change in sign of the \(d_K\) occurs, \(\tau _I\) is traversed by the immersed boundary \(\Gamma \).
At the acute corner in the figure, the region of points whose closest point is the vertex B, is delimited by the outer cone. For all points in this cone, \(\sigma _1\) and \(\sigma _2\) are possible choices as closest surface element. The cone contains the region ‘a’ in which the points are all outside with respect to both elements. The points in region ’b’ are outside with respect to one of the possible closest surface elements and inside with respect to the other. Hence, for this region the mentioned ambiguity can occur. One solution to this problem is to introduce angleweighted vertex normal vectors [26], but this requires extra data structures. Here we choose the simpler approach shown in Fig. 5: the point \(\varvec{x}_K\) has a larger distance to the extension plane of \(\sigma _2\) than to the extension plane of \(\sigma _1\). This distance is given by the inner product of the element normal vector and the distance vector between the considered grid point and the closest surface point (here, B). Choosing the element with a larger value of this distance resolves the ambiguity. The method is also used in three dimensions with the only difference being a larger set of candidates as closest elements.
Finally, we consider the numerical complexity of the distance function computation. If there are \(N_\Gamma \) elements in the surface mesh and \(N_\Omega \) nodes in the volume mesh, a bruteforce approach requires \(N_\Gamma \times N_\Omega \) closest point computations. In many cases, this number can be substantially reduced by precomputing a bounding box [32] of the surface \(\Gamma ^h\) and assigning a default value for the \(d_K\) of nodes outside of this box, but the essential complexity remains of order \(\mathcal {O}(N_\Gamma \times N_\Omega )\). Complexity reduction is possible by generation of a hierarchy of bounding boxes [32] or using socalled marching methods, see e.g. [33].
Constructive solid geometry modelling
The conventional finite element approach is to work through such a CSG pipeline, export a geometry representation and use a mesh generation software to create a bodyfitted volume mesh for the numerical analysis. The direct modification for an immersed finite element method is to export a surface representation of the geometry and embed this into the mesh by the methods described in “Immersed finite element method” section. Here, a third way is suggested in which the set operations are directly applied to the embedding (nonconforming) volume mesh. As outlined above, it suffice to provide the complement and intersection operations only. The former is trivially achieved: the use of a signed distance function generates an in and an outside partition of the mesh, reversing these partitions gives the complement. For this reason, all that need be explained is the intersection operation.
In the second embedding step, the distance function \({{\mathrm{dist}}}_{\Gamma _2}\) is used which gives rise to the element sets \(\mathbb {I}_2\), \(\mathbb {O}_2\) and \(\mathbb {C}_2\). All elements which belong to the outside are directly assigned to the complementary domain \(\Omega _\square {\setminus }\Omega \), that is \(\mathbb {O} = \mathbb {O}_1 \cup \mathbb {O}_2\). On the other hand, all elements of \(\mathbb {I}_1\), which also belong to \(\mathbb {I}_2\), are inside the final domain \(\Omega \), hence \(\mathbb {I} = \mathbb {I}_1 \cap \mathbb {I}_2 = \{\tau _{20}\}\). Finally, there are the intersection cases. Elements belonging to \(\mathbb {C}_1\) and \(\mathbb {I}_2\) (\(\tau _{21}\)) keep their status and subdivision. Elements from \(\mathbb {I}_1\) and \(\mathbb {C}_2\) (\(\tau _{10}\)) are subject to the same decomposition methods as \(\mathbb {C}_1\). It remains to discuss the situation of the elements which belong to \(\mathbb {C}_1 \cap \mathbb {C}_2\); the ones which are intersected by both boundaries, and in our example of Fig. 7 this is the element \(\tau _{11}\). In this case, simply the composing triangles are intersected with \(\Gamma _2\) as if they were elements of their own. Proper categorisation of these simplex shapes defines the final domain \(\Omega \) and its complement \(\Omega _\square {\setminus }\Omega \), see the middle picture of Fig. 7.
The advantage of this approach becomes clear when looking at the right picture of Fig. 7. Shown is the result for the same target domain \(\Omega \), but first the composition of the individual distance functions \({{\mathrm{dist}}}_{\Gamma _i}\) is computed according to expression (26) and then the element intersections are constructed. Clearly, in the right picture the corner is chamfered whereas in the above outlined approach this geometric feature is preserved. This is the distinctive characteristic of the presented idea: by successively embedding the geometry primitives into the mesh, the sharp features at the primitive intersections are preserved. It is important to remark that the boundaries \(\Gamma _i\) have been represented exactly in this example, but this is solely owed to the fact that they are straight lines. In the more general situation of curved boundaries, they are again represented on the finite element mesh by piecewise linear simplex elements. But, even though these surrogate boundaries do not exactly reproduce the given geometry, the here presented approach still allows to represent corners or edges at the intersection locations of the original primitives which lie inside of the finite elements.
Alternative approaches for increasing the quality of implicit geometry representations in the vicinity of sharp features (such as edges and corners) exist. In [38] the operations of surface reconstruction by means of marching cubes and the distance function computation are combined in order to generate a socalled directed distance field allowing for a better resolution of surface features. On the other hand, enriched distance functions are presented in [39] where additional edge and vertex descriptors augment the distance geometry representation. Although both approaches are promising concepts in the context of immersed finite element methods, they are not further considered in this work.
We conclude this paragraph by noting that the here used tessellation techniques also help to construct numerical integration schemes for the elements that are traversed by the boundary or interface. The cut elements are general polytopes for which quadrature rules are not easily obtained. There are many techniques that address this problem, such as momentfitting [40], surfaceonly integration [41], and adaptive decomposition of the integration region [14, 42]. But since we have a tessellation in simplex shapes already available, we use composite Gauß type quadrature rules, see e.g., [11].
Numerical stability
Up to now, it has been shown how to derive an immersed finite element method for boundary value and interface problems, see (9) and (19), and how to compute the matrix coefficients of the linear system of equations. But the stable solution of this final system of Eq. (21) remains to be discussed, especially in view of the method’s parameters \(\gamma \) (for boundary value and interface problems) and \(\beta \) (for interface problems only).
Sources of instability
In the case of the interface problems and formulation (19), the situation is slightly better. The extra parameter \(\beta \) can be adjusted in a smart way such that a finite value of \(\gamma \) is always achievable. Such a choice is proposed in [43] where \(\beta \) depends on the material parameters of the subdomains and the sizes of the cut elements, \(\tau _I \cap \Omega _i\). Using this approach, the system matrix has always positive eigenvalues (for the considered problem class) with a finite value of \(\gamma \). Nevertheless, the minimal eigenvalue goes to zero for vanishing sizes of the cut elements. Even though the parameter choices by [43] show a good performance in terms of the quality of the numerical results, the matrix condition number still cannot be bounded for a fixed mesh and arbitrary interface locations.
Stabilisation
 S1
Discarding all degrees of freedom with support intersection below a certain threshold, \(s_I < \varepsilon h\);
 S2
Adding a facebased stabilisation term [16];
 S3
Another approach, S2, is proposed in [16] where the jump of the function gradients across certain element faces is added to the weak form in order to guarantee stability of the method. Other than the result (31), the system matrix stays wellconditioned for small values of \(\gamma \) in the limit \(\varepsilon \rightarrow 0\). This approach requires to evaluate surface integrals over interior mesh faces, a technicality which requires additional data structures in many codes, but does not add much to the overall difficulty of implementing an immersed finite element method. Nevertheless there is a drawback with this approach, since it introduces another weighting factor whose adjustment is not straightforward: for too small values of this factor the stabilisation effect disappears and for too large values the method’s accuracy is affected [16].
There are two open questions when using this approach: (i) the choice of the index set \(\mathbb {A}(J)\) associated to J and (ii) the values of the constraint weights \(c_{IJ}\). The origin of this approach, as introduced in [12], is to stabilise bspline discretisations. In this particular situation, the underlying mesh is logically Cartesian and an explicit expression of the coefficients \(c_{IJ}\) can be given as a function of the multiindices used to label that grid. See also [6] for a more intuitive interpretation of the arising extrapolation of Lagrange polynomials and its efficient implementation. The aim of this stabilisation procedure is to maintain the convergence order of the method and therefore to not lose the polynomial approximation quality of the approximation (20) due to the constraints (34). In other words, the modified basis functions \(\tilde{\varphi }_I\) introduced in (35) have to represent the same polynomials as the \(\varphi _I\) themselves.
 1
 2for all \(J \in \mathbb {B}\)

find \(\tau _{K(J)}\) with all degrees of freedom from \(\mathbb {A}\) that is close to \(\varvec{x}_J\),

define the constraint coefficients as \(c_{IJ} = \varphi _I(\varvec{x}_J)\) for all \(I \in \mathbb {A}(J)\)

 3
assemble the final system of equations using the constraint equations (34) applied to test and trial spaces
 4
after solving the global system, calculate the constrained degree of freedom \(\varvec{u}_J\) with \(J \in \mathbb {B}\) according to (34).
With respect to the implementation a few remarks have to be made. The code has to be able to search the elements in the neighbourhood of a given node. For instance, the element \(\tau _{K(J)}\) in Fig. 10 does not lie in the support of \(\varphi _J\) but in the ring of elements around that support. Theoretically, for very extreme shapes of \(\Gamma \) the nearest \(\tau _{K(J)}\) to \(\varvec{x}_J\) could lie far away, but here we assume that the mesh is fine enough such that there is always an element nearby. Cuspshaped domains are excluded from the onset. In addition, one has to evaluate the shape functions of \(\tau _{K(J)}\) at \(\varvec{x}_J\) and this requires to find first the reference coordinate \(\varvec{\xi }_J\) (outside of the reference element) such that the geometry representation of the chosen element represents \(\varvec{x}_J\) when evaluated at this coordinate, that is \(\varvec{x}_{K(J)} (\varvec{\xi }_J) = \varvec{x}_J\). Here we restrict ourselves to meshes in which all elements are an affine transformation of the reference element. Higherorder geometry representations of the volume mesh are excluded, but they are also not necessary since the mesh, by design of the immersed method, need not conform to the geometry of \(\Omega \).
Numerical examples
At last, a few numerical examples are presented in order to study and demonstrate the performance of the immersed finite element method as presented here. Unless indicated otherwise, the spatial discretisation of all problems is carried out with linear finite elements. As shown in the appendix, the Nitsche parameter is chosen as \(\gamma = \gamma _0 \tfrac{\alpha }{h}\) with the mesh width h, the representative material parameter \(\alpha \) and a dimensionless scalar \(\gamma _0\). The default choices for this parameter is \(\gamma _0 = 10\) and for interface problems the additional parameter is chosen as \(\beta = 0.5\). The threshold \(\hat{s}\) used to distinguish between the degree of freedom sets \(\mathbb {A}\) and \(\mathbb {B}\) in the stabilisation of “Stabilisation” section is set to the size of one element.
Convergence and robustness analysis
At first, the method’s performance under variation of various parameters is assessed. For this purpose, an essentially onedimensional Poisson problem is used as depicted in Fig. 11, left, with a forcing function \(f(\varvec{x}) = \alpha ^2 \sin (\alpha x_1)\) and \(\alpha = \frac{3\pi }{2 x_\delta }\). The resulting exact solution is then \(u(\varvec{x}) = \sin (\alpha x_1)\). The signed distance function is \({{\mathrm{dist}}}_\Gamma (\varvec{x}) = x_\delta  x_1\) and the boundary is represented exactly. At first, this problem is analysed using a structured mesh as shown in the figure. Figure 11, on the right, shows the analytic solution (solid black line) along the \(x_1\)axis and its derivative (dashed line) for a boundary location at \(x_\delta = \frac{6}{7}\). The approximations \(u^h\) and \(\partial _{1} u^h\) for a mesh with \(5\times 5\) elements (red) and a \(10\times 10\) element (blue) are also displayed. One can see that the numerical approximation \(u^h\) coincides with the analytic solution at the finite element nodes and, moreover, at the boundary location at \(x_\delta \).
At last, we consider the influence of the geometry representation. As outlined in “Constructive solid geometry modelling” section, we have to approaches available: the use of a signed distance function representing the entire embedded surface and the successive embedding of the geometry primitives that form the final model. For simplicity, consider a square that coincides on two of its edges with the mesh boundary whereas the other two are represented implicitly. Figure 17 shows the effect of the introduced two approaches in the left and middle images, respectively. Clearly, the upper right corner is chamfered off in the first approach, but represented exactly in the second. As a numerical problem we have chosen \(\Delta u = 1\) on a unit square subject to \(u=0\) on the lower and left boundaries and \( \partial u / \partial \varvec{n} = 0\) on the other two boundaries. An analytic solution to this problem is available, for instance in [45] in the context of Poiseuille flow in a rectangular channel. The right graph in Fig. 17 shows the convergence rates for the two types of geometry modelling. Clearly, optimal convergence rates are obtained for both cases. Nevertheless the exact representation of the corner leads to a much smoother outcome with lower approximation errors for coarse mesh sizes.
Meshembedded CSG
 1
intersection with a sphere of radius 0.65 and centred at (0.5, 0.5, 0.5), and
 2
successive subtraction of cylinders around the same centre with radius 0.3 and in the directions of the \(x_i\)coordinate axes.
Composite material
Conclusions
Immersed finite element methods, that do not rely on a bodyfitted mesh, are a promising alternative to conventional FEM for many applications. Especially in the case of complex threedimensional geometries, moving interfaces, or design optimisation such methods allow for more flexible geometry processing and remove the repeated interaction with mesh generation software. Here, we present an immersed FEM for the problem class of nonlinear elasticity, based on a weak incorporation of Dirichlet boundary conditions and interface conditions with Nitsche’s method, an implicit geometry representation and accurate integration of the arising cut elements. We place emphasis on the implementation details such as the robust computation of the signed distance function and quadrature by means of tessellation. A common pitfall of nonbodyfitted FEM, the loss of numerical stability in situations with degenerate function support, is analysed and we provide a stabilisation technique that is robust without affecting the convergence behaviour. Moreover, the choice of the parameters in the context of Nitsche’s method are thoroughly discussed.
We demonstrate a way to incorporate sharp features such as edges and vertices in our method by means of successively embedding the geometry primitives into the analysis mesh in a similar way as in constructive solid geometry modelling. Based on this idea, geometry modelling is directly integrated in the finite element analysis and there is no need for a mesh generation tool. The presented applications emphasise the potential of this approach, where large deformation analyses are carried out based on a trivial Cartesian background mesh.
A present shortcoming of the introduced approach is the restriction to linear approximation orders. Although the field approximation used in this FEM can be of arbitrary order, a gain in convergence order would be impeded by the geometry representation based on linear facets. In principle, the use of more accurate signed distance functions and the subsequent adaptation on the quadrature level to account for embedded higherorder surface representations is feasible.
Finally, we note that the presented method is ideally suited for the incorporation of hadaptivity. A combination of this immersed FEM with hierarchical refinement techniques as shown, for instance, in [47] would render a powerful analysis toolbox, which yields accurate numerical predictions based only on the input of geometry primitives.
Declarations
Author's contributions
The presented ideas are joint contribution of the three authors. TR implemented the numerical techniques and wrote the initial draft of the paper. JMGA and FC contributed to the revision of the draft. All authors read and approved the final manuscript.
Acknowledgements
This work was partially supported by the EPSRC (second author, Grant #EP/G008531/1), by the European Research Council (third author, Grant #ERC2012StG 306751), and by the Spanish Ministry of Economy and Competitiveness (third author, Grant #DPI201564221C21R).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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