Numerical integration of discontinuous functions: moment fitting and smart octree

Transcription

1 Noname manuscript No. (will be inserted by the editor) Numerical interation of discontinuous functions: moment fittin and smart octree Simeon Hubrich Paolo Di Stolfo László Kudela Stefan Kollmannsberer Ernst Rank Andreas Schröder Alexander Düster Received: date / Accepted: date Abstract A fast and simple rid eneration can be achieved by non-standard discretization methods where the mesh does not conform to the boundary or the internal interfaces of the problem. However, this simplification leads to discontinuous interands for intersected elements and, therefore, standard quadrature rules do not perform well anymore. Consequently, special methods are required for the numerical interation. To this end, we present two approaches to obtain quadrature rules for arbitrary domains. The first approach is based on an extension of the moment fittin method combined with an optimization stratey for the position and weihts of the quadrature points. In the second approach, we apply the smart octree, which enerates curved sub-cells for the interation mesh. To demonstrate the performance of the proposed methods, we consider several numerical examples, Simeon Hubrich hubrich@tuhh.de Alexander Düster alexander.duester@tuhh.de Numerical Structural Analysis with Application in Ship Technoloy (M-10), Hambur University of Technoloy, Am Schwarzenber- Campus 4 (C), Hambur, Germany Paolo Di Stolfo paolo.distolfo@sb.ac.at Andreas Schröder Andreas.Schroeder@sb.ac.at Department of Mathematics, University of Salzbur, Hellbrunner Str. 34, 5020 Salzbur, Austria László Kudela laszlo.kudela@tum.de Stefan Kollmannsberer kollmannsberer@bv.tum.de Ernst Rank rank@bv.tum.de Chair for Computation in Enineerin, Technical University of Munich, Arcisstr. 21, Munich, Germany showin that the methods lead to efficient quadrature rules, resultin in less interation points and in hih accuracy. 1 Introduction The mesh eneration process of the classical finite element method (FEM) can become labor-intensive or practically impossible when facin problems that exhibit a complex eometry or consist of eometrical features such as voids, material inclusions, or cracks. In the past few decades, novel discretization methods have therefore been introduced to simplify the mesh eneration. Such methods are, for example, the extended finite element method (XFEM) [1 4] and the eneralized finite element method (GFEM) [5 7] which are based on the partition of unit method (PUM) [8, 9], the fictitious domain technique [10 13], or the finite cell method (FCM) [14 16] which combines the fictitious domain approach with hih-order finite elements. The basic idea of these methods is to separate the approximation of the primary variables from the approximation of the eometry. In doin so, the mesh is not restricted to coincide with the boundary or internal interfaces of the problem under consideration, which is why it allows for a fast and simple mesh eneration usin structured meshes or Cartesian rids, for instance. However, this simplification in the mesh eneration comes at the expense of a more sophisticated application of boundary conditions [17 20], a bad conditionin of the equation system [21 25], the necessity of a local enrichment stratey [26 30], and a more elaborate numerical interation method to handle the discontinuous interands for elements that are intersected by the boundary or by internal interfaces [14 16, 31 34]. In this paper, we focus on the numerical interation of arbitrarily broken elements. To distinuish these elements which do not have to conform to the boundary or the internal

2 2 Simeon Hubrich et al. interfaces from classical finite elements, will hereinafter denote them as cells. In the context of structural mechanics, we are interested in the computation of the interals related to the mass and stiffness matrix as well as the body force vector. Thereby, standard quadrature rules are applied for non-broken cells which are not intersected by the boundary or by internal interfaces. These quadrature rules, however, do not perform well for broken cells which face discontinuous interals of the followin form Ω C α(x) f (x)dω. (1) In Eq. (1), Ω C denotes the cell domain, f (x) is a continuous and smooth function defined over Ω C, and α(x) is a function introducin a discontinuity. In the framework of the XFEM, α(x) defines the Heaviside function and, in the context of the FCM, α(x) is the indicator function which is one for points that are located within the physical domain, and zero otherwise. Since standard quadrature rules do not perform well for the numerical interation of discontinuous functions adaptive interation schemes are required. To this end, one way to compute these interals is to replace the discontinuous interand of the cell by an equivalent polynomial and then to perform a standard Gauss quadrature. This idea has been proposed by Ventura and Benvenuti in [35, 36] and has been extended for the application to the FCM by Abedian et al. [37]. Another commonly used scheme is to split the interal in Eq. (1) into two parts α(x) f (x)dω = Ω C α(x) f (x)dω + Ω A α(x) f (x)dω Ω B whereby the interals on the riht-hand side consist of functions that are smooth and continuous over the correspondin interation sub-domains Ω A and Ω B, respectively. Havin the discontinuous interal subdivided into two continuous interals, however, does not ensure a straihtforward application of the standard quadrature rules since the subdomains Ω A and Ω B exhibit an arbitrary topoloy. To perform the computation of the interals over these particular sub-domains, a local interation mesh is thus enerally required, consistin of elements in which standard quadrature rules can be applied [2 4, 7, 15, 32 34, 37 45]. In the course of this, broken cells are subdivided into sub-cells, followed by computin the interals over these particular subdomains. Since the mesh is only needed for interation purposes, the adjacent cells do not need to conform, and the mesh may thus obtain hanin nodes. Consequently, the accuracy of the interation methods based on local meshes is related to the approximation of the correspondin interation domain. For this reason, there exist different procedures to enerate these meshes, such as methods based on a (2) quadtree or an octree subdivision [15, 32, 34, 40], low-order tesselation usin trianles or tetrahedrons [3, 4, 46, 47], or applyin hih-order sub-cells [7, 38, 44, 45, 48 50]. The adaptive interation schemes based on spacetrees subdividin broken cells by a quadtree (in 2D) or octree in (in 3D) have the advantae that they are robust, fully automatic, and allow to control the error in interation. Moreover, their implementation is very simple. The disadvantae, on the other hand, is that for a quadrature rule of hih accuracy a fine sub-cell mesh is usually required, resultin in a hih number of interation points. This hih number of interation points, in turn, renders the numerical interation expensive. To overcome this problem, we present two methods. In the first method, we extend the method suested in [51] where individual quadrature rules are enerated for every broken cell by solvin the moment fittin equations [52 56]. For the computation of the moments, the interation domain is represented by a trianulated surface mesh and, thus, results in a hih accuracy of its eometry description when usin fine meshes. Solvin the moment fittin equation system is not an easy task since it is nonlinear in terms of the position of the interation points. For this reason, the method developed in [51] follows the idea of fixin the position of the interation points a priori, as suested by Mousavi et al. [52]. In [51, 56] an adaptive point distribution scheme has been implemented to this end, subdividin the broken cell uniformly into sub-cells before the required interation points are distributed randomly in every sub-cell that is completely located within the interation domain. In doin so, the nonlinear moment fittin equation system turns into a linear system which is then solved by applyin a linear leastsquares fit. The advantae of this method is that the number of interation points is reduced sinificantly as compared to the adaptive interation based on an octree subdivision. Moreover, the adaptive point distribution scheme ensures that all interation points are located within the interation domain which corresponds to the physical domain in the context of the FCM. The disadvantae, on the other hand, is that the interation points provided by the adaptive distribution scheme may lead to a bad conditionin of the moment fittin equation system and, thus, may decrease the accuracy of the enerated quadrature rule. This problem is even more pronounced for hih-order quadrature rules. In this contribution, we present two approaches to overcome this problem by optimizin the accuracy of the moment fittin quadrature rules where the interation points have to be located within the domain of the broken cell but are not restricted to the interation domain. In the context of the FCM, this means that the points may be located within the physical as well as the fictitious domain of a broken cell. In the first variant of the moment fittin method, we define the position of the interation points a priori. In doin so, we take advantae

3 Numerical interation of discontinuous functions: moment fittin and smart octree 3 of the position of the Gauss-Leendre points, which lead to a ood conditionin of the moment fittin equation system and, thus, result in quadrature rules of hih accuracy. In the second variant of the moment fittin, we enerate quadrature rules by solvin an appropriate optimization problem related to the nonlinear moment fittin equations. First investiations in this reard have been presented in [57] where an optimization problem of the moment fittin method is solved drawin on the procedure proposed by Ryu and Boyd [58]. Althouh efficient quadrature rules are obtained, resultin in a low number of interation points, solvin the optimization problem becomes expensive for hih-order quadrature rules. For this reason, in this paper, we formulate an alternative optimization problem which is based on the minimization of the residual of the moment fittin equation system. In doin so, the moment fittin results in efficient quadrature rules, yieldin optimized points and weihts where the points have to be located within the cell but are not restricted to the interation domain. In the second method, we perform the numerical interation based on the smart octree method presented in [45]. The smart octree subdivides the interation domain into subdomains with smooth interands (see Eq. (2)). Similar to the classical octree, it subdivides the cut cell into eiht octants, but uses node-relocation and hih-order polynomials for interation sub-cells. It thereby captures the most primitive topoloical cases directly at the first level of refinement and uses recursion only as a fallback option, until such a primitive case appears. The smart octree thereby inherits the positive features of the standard octree approach, such as robustness and minimal topoloical complexity. At the same time, it can approximate complex eometries with hih accuracy. Due to this fact, the smart octree does, in eneral, not require many levels of subdivision. This results in a very low number of interation points where all points are located within the interation domain. Thus, the interation can be carried out with hih accuracy entirely on the physical domain in the context of the FCM. The paper is structured as follows: At first, the individual numerical interation methods are introduced and explained in detail in Sect. 2. Next, in Sect. 3, we demonstrate the efficiency and accuracy of the proposed interation methods considerin different numerical examples. Here, we start off by investiatin the numerical interation of polynomials on different domains before demonstratin how they are applied to the finite cell method. Finally, Sect. 4 concludes the paper ivin a summary and an outlook for future works. 2 Numerical interation 2.1 Moment fittin This section ives a brief overview of the moment fittin method. For more detailed information, the reader is referred to [53, 54, 59, 60]. The idea of the moment fittin is to enerate a quadrature rule by solvin the moment fittin equations n f j (x i )w i = f j (x)dω, j = 1,...,m (3) i=1 Ω A where x i are the n interation points, w i are the n correspondin weihts, f j (x) are the m independent basis functions, and Ω A the domain of interest. In matrix notation the moment fittin equations reads f 1 (x 1 )... f 1 (x n ) w. 1 Ω A f 1 (x)dω..... =., (4) f m (x 1 )... f m (x n ) w n Ω A f m (x)dω and can be summarized as follows Aw = b (5) where A defines the coefficient matrix composed of the function evaluations of the basis at the interation points, w is the vector of the weihts, and b is the vector of the individual interals of the basis functions over the interation domain Ω A. The interals of b are also denoted as moments. Thus, to conclude, the moment fittin can be understood as a eneral approach to set up a quadrature rule for an arbitrary domain Ω A. Since we are interested in the interation of polynomials, we use the orthoonal Leendre polynomials for the basis, which span the three-dimensional tensor product space F = { L r (ξ )L s (η)l t (ζ ), r,s,t = 0,..., p q }. (6) In Eq. (6), p q defines the order of the basis functions that form the interands of the riht-hand side in Eq. (3) and (4). Different methods can be applied to compute these interals. On the one hand, the volume interals can be computed by applyin special quadrature rules that suit the topoloy of the broken cell, as presented in [40], or by usin space trees like the standard octree [15] or the smart octree [45], for instance. On the other hand, the volume interals can be transformed into surface interals utilizin the diverence theorem, as shown in [51]. In doin so, the interals of the riht-hand side read f j (x) dω = Ω A div j (x) dω = Ω A j (x) n(x) dγ (7) Γ A

4 4 Simeon Hubrich et al. where Γ A defines the surface of the interation domain Ω A and n(x) its normal vector, which points in outward direction. Further, the vector j (x) defines the anti-derivatives which are computed accordin to [54] as j (x) = 1 f j (x) dx f 3 j (x) dy. (8) f j (x) dz In order to perform the interation, a parameterization of the surface is required. A rather simple approach to do so is to trianulate the surface and to apply a Gaussian quadrature on the trianles of the discretized surface [15]. Solvin the moment fittin equation system is not straihtforward since it is linear in the weihts and nonlinear in the position of the interation points. In [57], a nonlinear optimization approach followin the method suested by Ryu et al. [58] has been applied to enerate efficient quadrature rules usin optimized interation points. Althouh it could be shown that this method is very efficient usin less interation points than the number of the basis functions it proved to become expensive when computin quadrature rules of hiher order. A remedy to solve the nonlinear moment fittin equations is to follow the approach presented in [51, 53, 54, 56, 59]. To this end, the position of the interation points is selected in advance, thus transformin the nonlinear system into a linear one. In doin so, the number of the interation points is chosen to be equal or hiher than the number of the basis functions (n m), and the linear equation system is solved usin a linear least-squares fit. In [51, 56], the position of the points is selected by applyin an adaptive point distribution scheme where the broken cells are uniformly subdivided. Then, the interation points are distributed randomly over those sub-cells that are completely located within the physical domain. The advantae of this distribution scheme is that all points are located within the physical domain. The disadvantae, on the other hand, is that the moment fittin equation system becomes ill-conditioned for hih-order quadrature rules, resultin in a lower accuracy. In [51], it is shown that the standard weihts of the Gauss- Leendre quadrature can be recovered by applyin the moment fittin to a non-broken cell a cell that is not cut by the physical boundary when selectin the position of the Gaussian points. In doin so, even the weihts of the hihorder quadrature rules can be computed within machine precision. Consequently, usin the Gaussian points is a ood choice also for broken cells because the left-hand side in Eq. (3) and (4) will be the same as for the non-broken cell. This, in turn, means that the condition number of the system remains the same as well, thus ensurin hih accuracy for the hih-order quadrature rules. Moreover, applyin the standard Gaussian points to the moment fittin equation system transforms it into a linear square system so that the number of the interations points equals the number of the basis function (n = m). Althouh this implies allowin the points to lie within the fictitious domain, we can show that this approach is suitable to consider linear problems of the finite cell method. 2.2 Optimization of quadrature points and weihts We denote by P k the set of polynomials of deree up to k on [ 1,1]. In this one-dimensional settin, it is well-known that, with n Gauss-Leendre points and weihts, each polynomial in P 2n 1 can be interated exactly. Also, it can easily be shown that there are no n points and weihts, so that each polynomial in P 2n can be interated exactly. Therefore, the choice of points and weihts in the Gauss-Leendre quadrature is optimal in the sense that each polynomial in P k can be interated exactly for the hihest deree k possible when n points are used. However, the situation in hiher dimensions is sinificantly more complex than the one-dimensional one, mainly due to the fact that domains in hiher dimensions come in a far reater variety of shapes [58, 61]. Thus, for eneral domains in hiher dimensions, there are no eneral rules to determine points or weihts. Consider the moment fittin system (5) for m basis functions and n = m = (p q +1) 3 points. If the points are pairwise distinct, the system is invertible, implyin that there is a set of m points, such that each basis function in F can be interated exactly. Note that, if we applied the same rule for the number of quadrature points as in one dimension to interate the set of m functions exactly, a Gauss-type quadrature formula in 3D would require 1 2 ( pq + 1 ) 3 points, i.e., only approximately 12.5% of the oriinal number of points. Especially in the context of nonlinear problems, where a repeated computation of expensive interals (such as the stiffness matrix) is required, even a sliht reduction of the number of quadrature points can improve the computational complexity of the whole computation in a sinificant manner. Therefore, it is of reat interest to numerically determine cheaper quadrature rules, i.e. rules with fewer points, by which a set of basis functions can still be interated exactly. A eneral approach aimin at the reduction of the number of points and weihts n serves to solve an optimization problem related to the moment fittin idea: For a iven vector b of interals of a set of m basis functions, we aim to find the least number n and a set of n points and weihts, such that the residual r 2 2 = r(x 1,...,x n ) 2 2 := b A(x 1,...,x n )w(x 1,...,x n ) 2 2 vanishes, where A(x 1,...,x n ) denotes the moment fittin matrix and w(x 1,...,x n ) denotes the solution of the moment

5 Numerical interation of discontinuous functions: moment fittin and smart octree 5 fittin system (5). More precisely, our stratey aims at findin points x 1,...,x n with minimal n < m such that r 2 is count for the more detailed surface information available kind of eometric representation. However, it does not ac- reduced to ε mach, where ε mach is the machine precision. We from the eometric model. Therefore, in order to increase note that the numerical value of r 2 heavily depends on the accuracy of the interation, many levels of octree refinement may be required. This results in a hih number the condition number of the matrix A. Since A is invertible and well-conditioned when the set of m = ( p q + 1 ) 3 tensorproduct Gauss-Leendre points is used, we et the trivial overcome this issue, the method of smart octrees has been of quadrature points, makin the simulation expensive. To upper bound n m, which implies that the minimization recently introduced in [45]. This alorithm combines the robust features of the octree procedure with an automatic res- problem has a solution. As the problem belons to the nonlinear least-squares olution of the intersection topoloy includin sharp edes minimization problem class, several well-established iterative techniques are available for its solution. A well-known surface representation. In the followin, the key aspects of and corners that may be present in the cell and hih-order alorithm is the Levenber Marquardt alorithm that, essentially, performs a combination of the Gauss Newton method The standard octree refinement enerates eiht octants in the the alorithm are presented. and a radient descent. Another alorithm, which can also cut cells by eneratin 19 new internal nodes, which can be be applied for more eneral problems, is the Sequential Quadraticcateorized as follows: Prorammin (SQP) alorithm. A major shortcomin of the proposed method is its computational performance. The first performance issue stems from the fact that the minimal number n of points is not known. More importantly, it is unknown whether, for fixed n < m, the function r 2 has a lobal optimum; also, the structure and distribution of local minima is unknown. Therefore, we have to estimate n. Since the residual r 2 reaches machine precision for n = m, we may assume that r 2 decreases when the number of points n is increased. Thus, we may assume that there is a number n, and r 2 attains machine precision for some n points, while r 2 does not attain machine precision for any n 1 points. Therefore, we may apply a binary search method that starts with 0 and m points to find the optimal number n in approximately lo(m) executions of the optimization method. A second issue addresses the performance of each execution of the underlyin optimization method. The methods are usually sensitive to the choice of initial points. Suboptimal initial points may result in a slow decrease of the error, resultin in a hih number of iterations. Thus, it is necessary to set a reasonable limit to the number of iterations. Even for a ood choice of initial points, the method may pursue a local minimum that is far off from the optimum. As the structure of the domain of interation is unknown to the method and the computation is performed on the reference element [ 1,1] 3, it is safest to initially distribute n points at random. We note that, in order to assess the actual minimal number n of points for iven b with hih probability, several runs of the binary search method have to be performed. ede nodes, lyin on the bisection point of the cell edes, face nodes, lyin on the center of ravity of each cell face, a mid-node, created at the center of ravity of the cell. By construction, the octree procedure distributes the internal nodes to pre-defined locations, the center of ravity of the edes, faces, and the cell. The key idea of the smart octree method is to relay this constraint. The internal nodes can be moved around freely, as lon as they stay on the respective components of the cell (e.. an ede node remains on an ede). Therefore, before distributin the internal nodes, the method examines the components of the cell and cateorizes them as follows: If an ede is intersected by the interface, it is identified as an active ede. The correspondin ede node is moved onto the intersection point. If a face is intersected by the interface, it is identified as an active face. For a iven active face, the face node is shifted onto the intersection curve between this face and the interface. Finally, if the cell contains an active element, the mid-node is shifted onto the interface while remainin inside the cell. The concept of distributin internal nodes is depicted in Fi Smart octree As mentioned in the introduction, the advantae of the classical octree-based interation is that it works robustly on any Fi. 1: Smart octree eneration. The nodes on active edes, faces, and the mid-node are moved onto the interface. For simplicity, only two resultin sub-cells are shown [45]

6 6 Simeon Hubrich et al. While the location of active ede nodes becomes completely constrained by the intersection point, the active face nodes and mid-nodes have remainin derees of freedom. For example, an active face node needs to lie on the intersection curve between the interface and the face, but can be moved alon this intersection curve freely. Further, an active mid-node can be shifted to any location on the interface, as lon as it remains inside the cell. The freedom in movin the active face nodes and the mid-node becomes useful when the cell contains sharp edes or corners. Consider the situation (also depicted in Fi. 2) of a sharp ede e enterin the cell throuh one of its faces f. In this case, the active face node on f is constrained to lie on the intersection point e f, therefore its location becomes completely prescribed. Further, the constraint on the location of the mid-node becomes stricter: it has to lie not only on the interface, but also on the sharp ede e. However, alon this ede e, it can still be shifted freely. The last re- The outlined method requires that each active ede, face, and cell can uniquely be associated to a sinle internal node. If this requirement is violated, the smart octree method subdivides the cell into eiht equal octants by a simple octree refinement step. The alorithm then attempts to resolve the interface in each of the resultin sub cells. This octree-like refinement step is repeated recursively until the criterion of a sinle internal node per ede/face/cell is fulfilled. If a maximum number of refinement levels k max is reached, and the cut case can still not be resolved by smart octree decomposition, the fallback stratey is to compute interals by classical octree. The outlined steps of the smart octree method partition the cut cells into eiht octants, where the domain interface is approximated linearly. If the eometric model is composed of hih-order entities, the trilinear cells are reparametrized usin the method of blendin function interpolation [62] in order to conform with these curved boundaries. To this end, a local reparametrization of the interface is applied by interpolatin a set of sample points projected onto the interface. The sample points are interpolated usin Laranian shape functions. In Section 3.1.3, it will be demonstrated that the accuracy of the numerical interation based on smart octrees is controlled by the polynomial order p B of the interpolatin polynomials. Fi. 2: Sharp ede resolution. The active surface nodes (blue) lie at the intersection between the sharp edes (reen color) and the respective face. The mid-node (purple) can be placed on the sharp ede arbitrarily [45] mainin deree of freedom of the mid-node becomes completely constrained when the interface inside the cell contains a corner v. In this case, the active ede- and face nodes are distributed similarly as before, while the location of the mid-node is prescribed by the location of v. Fi. 3 depicts an example where the interface cuttin throuh the cell contains a corner and three sharp edes. 3 Numerical examples In this section, we discuss the performance of the proposed interation methods. To this end, we start off by investiatin the accuracy in the interation of polynomial interands for broken cells. Next, we combine the interation methods with the FCM and study the accuracy and efficiency considerin problems of different complexity. 3.1 Cell cut by a sphere In the first example, we contemplate a problem with an interation domain includin a curved surface a cell that is cut by a sphere. In doin so, we consider a hexahedral cell of the followin domain Ω C = [0, 1] 3. (9) The eometry of the sphere is iven by the followin level set function Fi. 3: Sharp corner resolution. The mid-node of the cell is moved onto the sharp corner, while the active surface nodes lie at the intersection between the sharp edes (reen color) and the respective face [45] φ (x) = (x x c ) 2 + (y y c ) 2 + (z z c ) 2 r 2, (10) where the center coordinates x c, y c and z c of the sphere and its radius r are iven as x c = y c = z c = 0 and r = 1. (11)

7 Numerical interation of discontinuous functions: moment fittin and smart octree 7 Thus, the interation domain of the broken cell is defined as Ω A = {x Ω C : φ (x) 0}. (12) Consequently, the eometry of the interation domain describes an eihth of a sphere, as depicted in Fi. 4. is plotted in Fi. 5. As it can be seen from the fiure, the GLP yield a much better conditionin than the APD. Applyin the GLP improves the conditionin of the moment fittin equations system sinificantly. Moreover, usin the GLP leads to the same conditionin of the system for any problem under consideration since the left-hand side in Eq. (4) is independent of the interation domain and only depends on the position of the interation points and the predefined basis functions. The conditionin for the APD, on the other hand, varies from problem to problem since the position of the interation points depends on the topoloy of the broken cell APD GLP Fi. 4: Geometry of the interation domain of the broken cell cut by a sphere condition number κ [-] order of the quadrature p q [-] Moment fittin To enerate quadrature rules that are exact up to a certain order, the interals of the riht-hand side in Eq. (4) have to be computed exactly. Since the interation domain is described by an eihth of a sphere, neither the quadrature of the rihthand side of Eq. (4) as a volume interal applyin an octree scheme or a surface interal based on trianulation will be exact. Due to this fact, we computed the interals symbolically usin Wolfram Mathematica [63]. An appropriate choice for the position of the interation points is required to provide a ood conditionin of the matrix on the lefthand side in Eq. (4). On this account, we study the condition number of the moment fittin equation system applyin the points provided by the adaptive point distribution (APD) scheme provided in [51] and compare it to those usin the position of the standard Gauss-Leendre points (GLP). To this end, we compute the condition number κ for different quadrature orders p q = 0,...,16 which are required when applyin hih-order shape functions in the FCM. Here, κ of A is defined as κ = σ max σ min, (13) where σ max and σ min are the maximum and minimum sinular values of the system, computed by the LAPACK routine DGELSS [64]. The condition number of both point sets Fi. 5: Condition number of the moment fittin equation system applyin different quadrature orders Due to the better conditionin of the moment fittin equation system usin the GLP, the solvability of the system is improved sinificantly. Next, to point out this fact, we consider the L 2 -norm of the residual r 2 of the moment fittin equations where the residual is defined as r j = f j (x)dω Ω A or, in matrix notation, n i=1 f j (x i )w i, j = 1,...,m (14) 1 Ω A f 1 (x)dω f 1 (x 1 )... f 1 (x n ) w r. = , r m Ω A f m (x)dω f m (x 1 )... f m (x n ) w n (15) where r j are the individual components of the residual vector of the system. Fi. 6 shows the L 2 -norm of the residual r 2 applyin different quadrature orders p q = 1,...,16 of the moment fittin. Here, it can be seen that for the application of the GLP the norm of the residual remains close to zero within machine precision for all quadrature orders,

8 8 Simeon Hubrich et al. while it appears to increase if the APD is used. Thus, we could reduce the norm of the residual by a factor within the rane between 10 4 and 10 5 for the hih-order quadrature rules (p q > 9). norm of the residual [-] APD GLP order of the quadrature p q [-] Fi. 6: Norm of the residual of the moment fittin equation system applyin different quadrature orders Next, we consider the conditionin of the moment fittin quadratures. A common definition for the condition number κ q of quadrature rules is [65] κ q = n i w i, (16) where w i are the individual weihts and n the number of interation points. To simplify the investiation of the conditionin, we normalize κ q by the volume V A of the interation domain Ω A κ q = κ q V A. (17) Fi. 7 shows the normalized condition number κ q of the moment fittin usin the APD and the GLP for quadratures of different order p q. From the fiure it can be seen that with increasin the order of the quadrature rule applyin the APD, κ q deviates sinificantly from the optimal value which is 1. This deviation is due to two reasons. Firstly, the moment fittin quadrature is composed of neative and positive weihts, and secondly, the individual weihts have a hih absolute value which is in particular the case for the hihorder quadratures. On the other hand, applyin the GLP for the moment fittin results in a ood conditionin for the quadrature rules althouh some of the weihts miht still be neative, which is evident from the fact that κ q slihtly oscillates between 1 and Finally, we investiate the accuracy of the enerated quadrature rules by interatin polynomials of order p i = 0,...,17. In doin so, we consider four different orders of the moment normalized condition number κ q [-] APD GLP order of the quadrature p q [-] Fi. 7: Condition number of the moment fittin quadratures applyin different quadrature orders fittin quadrature p q = 4,7,11,16. Since the interals of the riht-hand side in Eq. (4) are computed symbolically usin Wolfram Mathematica, the numerical interation has to be exact for all polynomials where the order of the polynomial is less or equal to the order of the quadrature (p i p q ). To measure the accuracy, we consider the relative error e r in interation, which is defined as e r = I ex I q I ex, (18) where I q is the value of the interal obtained with the moment fittin quadrature, and I ex is the exact value of the interal that is computed symbolically usin Wolfram Mathematica. The results of the APD and the GLP are plotted in Fi. 8 and 9, respectively. From the fiures, it can be seen that the relative error oscillates around a certain error limit for p i p q and that a lare jump occurs if p i exceeds p q. However, the error level of the APD is sinificantly hiher as for the GLP, where the error is close to zero within machine precision. This stron deviation when applyin the APD is due to the hih values for the norm of the residual of the moment fittin equation system, which oriinates from the bad conditionin of the system. Ultimately, we can reduce the relative error by a factor up to 10 6 for p q = 16 by utilizin the GLP. For p q = 16, the points of the GLP are plotted in Fi. 10a, and the points of the APD scheme as well as the correspondin sub-cell mesh are plotted in Fi. 10b and Fi. 10c, respectively Optimization of quadrature points and weihts The performance of the optimization of quadrature points and weihts described in Sect. 2.2 is to be assessed in two aspects. Firstly, we want to investiate the optimal number n of quadrature points that lead to a residual r 2 in the manitude of the machine precision for the cell cut by a sphere.

9 Numerical interation of discontinuous functions: moment fittin and smart octree 9 relative error e r [-] APD - p q =4 APD - p q =7 APD - p q = APD - p q = polynomial order of the interand p i [-] Fi. 8: Relative error in interatin polynomials with the moment fittin usin the APD relative error e r [-] GLP - p q =4 GLP - p q =7 GLP - p q =11 GLP - p q = polynomial order of the interand p i [-] Fi. 9: Relative error in interatin polynomials with the moment fittin usin the GLP Secondly, we aim to check for accuracy by measurin the relative error in the numerical interation of the same trial polynomials as in Sect , but usin the quadrature points and weihts determined by the optimization alorithm. For the first aspect, we apply the binary search method described in Sect. 2.2, which aims to find an approximation to the optimal number of points n such that the residual attains machine precision, but does not attain machine precision for any n 1 points. As the optimization alorithm ets hihly time-consumin for larer n and p q, we only try to find the optimal number n for quadrature orders p q = 2,...,7. The results are listed in Table 1. The results indicate that the number of points may be reduced sinificantly compared to the standard Gauss-Leendre quadrature by applyin the optimization alorithm. With an increase in the order of the quadrature, however, the ratio of n to the number of Gauss-Leendre points ( p q + 1 ) 3 becomes poorer in favor of the GLP. It can be seen that the final residual almost reaches machine precision for each p q. Quadrature order Optimal n by alorithm Residual r (< 27) (< 64) (< 125) (< 216) (< 343) (< 512) Table 1: Comparison of the least number n of points required for the residual r 2 to reach machine precision in comparison to the number of quadrature points for the Gauss- Leendre quadrature To assess the accuracy of the quadrature formula oriinatin from the points and weihts determined by the optimization alorithm, we measure the error that occurs when the trial polynomials from Sect are interated. The results for interands of polynomial order p i = 2,...,7 are depicted in Fi. 11. The results show that the relative error remains stable in the rane of to for all p i = 2,...,7. Interands of hiher deree than the order p q cannot be interated with sufficient precision Smart octree For comparison, the 18 different polynomial interands were also interated usin the smart octree decomposition alorithm, which was discussed in Section 2.3. Because each ede and face contains a sinle active internal node, the alorithm is able to decompose the cut cell into octants without havin to perform any additional octree refinements. Since the cell is cut by a curved eometry, the hih-order reparameterization of the sub-cell faces needs to be employed. Fi. 12 shows the resultin interation sub-cells of the smart octree decomposition. As mentioned in Section 2.3, the accuracy of the interation is determined by the polynomial order of the surface interpolation p B. An example is depicted in Fi. 13, where a 14 th order polynomial is interated by a 20 point quadrature rule, on interation cells with differin p B. As the polynomial order of the surface approximation increases, the error of the numerical interation drops rapidly. This relationship can also be observed in Fi. 14 and 15, where the error curves of the polynomial interands are depicted, for p B = 4 and p B = 12, respectively. For a lower quality approximation, such as p B = 4, the error for different polynomial interands is in the rane of For a sinificantly better surface approximation, these error levels can be drastically reduced, as depicted in Fi. 15 for p B = 12. Because the order of the mappin of the interation cell increases as p B increases, the order of the quadra-

10 10 Simeon Hubrich et al. (a) (b) (c) Fi. 10: Moment fittin quadrature for p q = 16. a GLP. b APD. c Correspondin sub-cell mesh of the APD relative error e r [-] Opt - p q =2 Opt - p q =3 Opt - p q =4 Opt - p q =5 Opt - p q =6 Opt - p q =7 ture rule needs to be increased as well. This can be observed in Fi. 15, where the curve representin p q = 4 is not able to reach the error levels of hiher order quadrature rules. Concernin the computational effort, even in cases of a very hih polynomial order p B, the increase in interation time of the cut elements is moderate p i =14;p q = polynomial order of the interand p i [-] Fi. 11: Relative error in interatin polynomials with points and weihts obtained by the optimization method relative error e r [-] polynomial order of surface interpolation p B [-] Fi. 13: Relative error of interation with smart octrees, dependin on the order of surface interpolation 3.2 Hydrostatic sphere Fi. 12: Smart octree interation mesh enerated on a sphere octant In the next example, we apply the proposed interation methods to a problem of structural mechanics utilizin the FCM [14 16] and compare the results to the adaptive interation based on an octree subdivision which is commonly used within the context of the FCM. To this end, we consider a sphere with a uniform traction applied to its surface in normal direction [15, 51, 66], as depicted in Fi. 16a. Thus, the

11 Numerical interation of discontinuous functions: moment fittin and smart octree 11 relative error e r [-] p q =4 p q =7 p q =11 p q = polynomial order of the interand p i [-] Fi. 14: Relative error in interatin polynomials with smart octrees, p B = 4 relative error e r [-] p q =4 p q =7 p q =11 p q = polynomial order of the interand p i [-] Fi. 15: Relative error in interatin polynomials with smart octrees, p B = 12 sphere is subjected to a hydrostatic stress state. For the analysis, the material behavior of the sphere is assumed to be isotropic linear elastic with Youn s Modulus E = 1.0 MPa and Poisson s ratio ν = 0.3. The radius of the sphere is R = 5.0mm, and the load on its surface is t n = 1.0MPa. Provided that the center of the sphere is fixed, the analytical solution of the displacement field of the linear problem reads Cx u(x) = Cy with C = 1 E (1 2ν) t n. (19) Cz Due to the symmetry of the problem under consideration, only one eihth of the sphere has to be taken into account. Thanks to the fictitious domain approach, the sphere is embedded into a fictitious domain, resultin in a domain of an octant which can be easily discretized. Since the solution is smooth and the FCM separates the approximation of the displacement field from the approximation of the eometry, only one finite cell is needed for the discretization, see Fi. 16b. Due to the linearity of the displacement field, an Ansatz of order p = 1 is sufficient to solve the problem exactly. This example presents a problem that is similar to a patch test, where a homoeneous stress field is considered. Usin only one cell within a fictitious domain approach allows to obtain the exact solution, provided that the eometry is treated appropriately and the computation of the stiffness matrix and load vector is consistent. Fi. 16b shows the model of the FCM analysis. Here, symmetric boundary conditions are applied to the cell by fixin its face to the bottom in z-direction, its face on the left-hand side in y-direction, and its face on the back in x-direction while the load is applied to the curved surface of the physical domain which is located within the cell. As it can be seen in Fi. 16b, the cell is subdivided into two domains. Here, only the physical domain defined by the sphere is of interest for the computation of the stiffness matrix. Consequently, the physical domain has to be resolved durin the analysis. In the framework of the FCM, this circumstance is taken into account by introducin the indicator function { 1 if φ(x) 0 α(x) = (20) 0 otherwise where the domain of the sphere is implicitly defined by followin level set function φ(x) = x 2 + y 2 + z 2 r 2. (21) Havin defined the indicator function α(x), the physical domain can be resolved very easily by a suitable interation mesh. In the case of the presented moment fittin methods, the physical domain is needed for the computation of the interals of the riht-hand side. To this end, the surface coverin the eihth of the sphere is iven by a trianulated surface mesh consistin of 525, 718 trianles. The surface mesh is depicted in Fi. 16c. Computin the volume, the relative error results in a value of about As can be inferred from Fi. 17, hierarchical refinements are required to obtain the same error level applyin the adaptive interation based on an octree subdivision 8. Here, the error in volume is plotted over the total number of interation points, applyin a Gaussian quadrature of order p q = 2 on every sub-cell. The correspondin sub-cell mesh of 8 hierarchical octree refinements is depicted in Fi. 16d, resultin in a number of 2,061,424 interation points. For the application of the inhomoeneous Neumann boundary conditions, a parametric description for the curved part of an eihth of the sphere is needed where the load acts in normal direction. To this end in the case of the adaptive interation and the moment fittin we use the trianular

12 12 Simeon Hubrich et al. t n = 1MPa (a) z y x (b) (c) (d) Fi. 16: Sphere under hydrostatic stress state. a Setup of the problem. b The FCM model usin one finite cell. c Trianulated surface of the eihth of the sphere. d Octree mesh with 8 refinements. relative error in volume [-] Octree refinement total number of interation points n [-] Fi. 17: Relative error in volume applyin the adaptive interation based on an octree subdivision for p q = 2 mesh depicted in Fi. 16c and only take trianles into account that describe the curved part of the sphere s surface. Havin the parametric description of the surface, the correspondin load vector is computed performin a numerical interation on every trianle as described in [15]. Next, we study the efficiency of the proposed interation methods by considerin the error in enery norm and the von Mises stress σ vm as the major quantities. Since the sphere experiences a pure hydrostatic stress state, the stress deviator vanishes and, thus, the von Mises stress does as well. Fi. 18 shows the contour plots of the von Mises stress for the adaptive interation, the smart octree approach, and for the moment fittin based on a trianulated surface mesh and on an octree mesh for the interation of the moments. For the smart octree approach, the order of surface interpolation was chosen as p B = 4, and (p + 5) 3 interation points were distributed in each interation cell. Here, it can be seen that the moment fittin usin a surface mesh and the smart octree result in a hiher accuracy than the adaptive interation and the moment fittin based on an octree interation of the moments. Usin the same discretization for the interation of the stiffness matrix as for the interation of the load vector as it is the case for the moment fittin usin the trianulated surface mesh and the smart octree results in a hiher accuracy. This fact is demonstrated min more detail by considerin the error in the von Mises stress. To this end, the relative error in the von Mises stress is defined as e vm = σ vm t n 100[%]. (22) For the investiation, we plot the error in the von Mises stress alon a radial line x = y = z [0, R/ 3] as depicted in Fi. 19 where APD, GLP, and OP are the results of the moment fittin usin a trianulated surface mesh and GLP OT denotes the results of the moment fittin utilizin an octree mesh for the computation of the moments. It can be seen that the results of the moment fittin based on a surface mesh are almost identical with the smart octree, whereas the results of the octree are sinificantly hiher. The results of the adaptive interation coincide with the results of the moment fittin if the same octree mesh is used for the computation of the moments. Since we have almost the same error level in the numerical interation, this difference oriinates from the different discretization of the mesh used for the computation of the stiffness matrix and the mesh used for the computation of the load vector, as mentioned before. Consequently, a nice feature of the moment fittin is that the same discretization can be applied for the computation of the load and the stiffness matrix. This feature also holds for the smart octree approach, because the resultin interation cells are boundary conformin. Therefore, those faces of the interation cells that lie on the interface can be directly used for the application of the boundary conditions. In the followin, we consider the relative error in enery norm as an additional quantity which is defined as e E(Ω) = U ex U FCM U ref 100%. (23)

13 Numerical interation of discontinuous functions: moment fittin and smart octree 13 (a) (b) Fi. 18: The von Mises stress for the sphere under hydrostatic stress state. a Adaptive interation based on 8 octree refinements and moment fittin usin the same adaptive interation scheme for the computation of the moments. b Smart octree and moment fittin for the trianulated surface mesh usin APD, GLP, and OP relative error in von Mises stress [%] octree interation moment fittin - APD moment fittin - GLP moment fittin - OP moment fittin - GLP OT smart octree - p b = R [mm] Fi. 19: Relative error in the von Mises stress alon the diaonal cutline In Eq. (23), U ex is the analytical value of the strain enery which is defined as U ex = 1 3C t n dω = π [J] [J] (24) and U FCM is the strain enery of the FCM analysis. U FCM is computed as U FCM = 1 2 UT KU, (25) where U is the displacement vector and K denotes the stiffness matrix. The values of the relative error in the enery norm of the different interation methods are listed in Tab. 2 where the superscript OT denotes the octree, SOT the smart octree, APD the moment fittin usin the adaptive point distribution, GLP the moment fittin utilizin the position of the Gauss-Leendre points, GLP OT the moment fittin with position of the Gauss points applyin an octree mesh for the computation of the moments, and OP the moment fittin with optimized position of the points and the weihts. Here, it can be seen that the relative error is almost the same with a value of about 0.2 [%] for the moment fittin and the adaptive interation based on an octree. This is due to the fact that these interation methods have approximately the same error level in the volume interation. The lower value of the smart octree is due to the hiher accuracy of the eometry approximation. Table 2: Relative error in enery norm iven in percentae p e OT E(Ω), e GLPOT E(Ω) e APD E(Ω), e GLP E(Ω), e OP E(Ω) e SOT E(Ω) [%] 0.2 [%] 0.07 [%] Considerin the number of interation points, n emphasizes the main advantae of the presented interation methods in comparison to the standard adaptive interation based

14 14 Simeon Hubrich et al. (a) (b) (c) (d) (e) Fi. 20: Position of the interation points for the sphere under hydrostatic stress state. a Adaptive interation based on 8 octree refinements. b Moment fittin usin APD. c Moment fittin usin GLP. d Moment fittin usin OP. e Adaptive interation based on a smart octree usin p B = 4 for the surface interpolation on an octree. To this end, the total number of interation points of all methods are listed in Tab. 3. As it can be inferred from the table, the moment fittin reduces the number of interation points by a factor of about Moreover, utilizin optimized points for the moment fittin reduces the number of interation points by a factor of about 3. The points of the different interation methods are illus- p Table 3: Total number of interation points n OT n APD, n GLP, n GLPOT n OP n SOT 1 2,061, trated in Fi. 20 with respect to the eometry of the cell and the topoloy of the physical domain. Here, it can be seen that the points of the adaptive interation and the moment fittin applyin the APD are located within the physical domain of the cell. For the moment fittin utilizin the GLP and OP, however, the points are located within the physical and within the fictitious domain of the broken cell. 3.3 Porous domain under pressure As the final example of this section, we investiate a problem of a porous material. In doin so, we consider a cube containin 27 ellipsoidal holes. The dimensions of the ellipsoids and their center are chosen randomly. The dimensions of the cube are iven as mm 3. Fi. 21a shows the eometry and the boundary conditions of the problem under investiation. As it can be seen from the fiure, the domain is subjected to symmetric boundary conditions the cube s face at the back is fixed in x-direction, its face at the left-hand side in y-direction, and its face at the bottom in z- direction. A uniform pressure of 100MPa is applied to the top face. The material behavior of the porous domain is assumed to be isotropic linear elastic where the Youn s modulus E is 5.0GPa and Poisson s ratio ν equals 0.3. For the FCM analysis, the domain is discretized with a Cartesian rid of cells. This results in a total number of 512 cells, of which 175 are broken. For the investiation, we compare the moment fittin to the adaptive interation based on an octree and a smart octree subdivision of the broken cells. In doin so, we define an error level of about 10 5 in the volume interation for all interation methods under consideration. In case of the moment fittin where we compute the moments via a surface interation we enerate a trianular surface mesh for the whole eometry, for which the error in the eometry description can be easily defined by any CAD proram. Since the mesh, however, represents the whole eometry, boolean operations have to be performed between the broken cells and the eometry. To this end, we use the Cork Boolean/CSG library [67] and perform intersections to obtain the individual surface meshes for the physical domain of the broken cells. The intersection between a broken cell and the eometry is demonstrated in Fi. 21c, just to ive an example. To obtain the same error in interation, applyin the adaptive interation based on an octree subdivision is not exactly straihtforward since the accuracy depends on the number of refinements and on the number of interation points of every sub-cell which depends on the order of the quadrature rule of the sub-cell. The order of the quadrature rule p q, on the other hand, depends on the order of the Ansatz p (p q = 2p). For this reason, we computed the error in the volume interation applyin different orders of the quadrature rule (p q = 2,4,8) for 1,2,3, and 4 refinements of the octree. The results are plotted in Fi. 22. From the fiure, it can be inferred that 3 octree refinements capture the error of about 10 5 in the volume interation for p q = 4,8. For p q = 2, however, the error in volume interation is about 10 4, even for 4 octree refinements. Due to this reason, we choose 4 octree refinements for adaptive interation, to be on the save side. The inside of the correspondin sub-cell mesh usin 4 octree refinements is exemplarily depicted in Fi. 21d. The interation mesh provided by the smart octree decomposition employs a hih-order reparametrization of the in-

15 Numerical interation of discontinuous functions: moment fittin and smart octree 15 t n = 100MPa z y x (a) (b) (c) (d) Fi. 21: Porous domain containin 27 ellipsoidal holes. a The FCM model. b Cartesian rid with c Trianular surface mesh of one broken cell used for the moment fittin. d Octree mesh with 4 refinements. terface, with p B = 4 for the polynomial order of the surface interpolation. Thus, the hih-order mappin may have to be taken into account by the order of the quadrature rule p q. To investiate the effects of p q on the accuracy of the FCM computations, we tested two scenarios, with p q = p + 1 and p q = p + 4. relative error in volume [-] Octree refinement p q = 2 p q = 4 p q = 8 Octree refinement Octree refinement total number of interation points n [-] Fi. 22: Relative error in volume applyin different orders p q for the quadrature rule of the sub-cells As a measure for the accuracy, we aain consider the relative error in enery norm, which is defined in Eq. (23) and replace U ex by U ref = J which is a reference solution obtained by an overkill FEM analysis. To study the converence behavior applyin the moment fittin and the adaptive octree interation, the relative error in enery norm is computed for different orders of the Ansatz p. In Fi. 23, the relative error in enery norm is plotted aainst the number of derees of freedom in a double loarithmic diaram. As it can be seen from the fiure, the converence behavior is almost the same. As mentioned before, this is due to the usae of approximately the same error level for all interation methods. However, the quadrature rules based on the optimization of the position of the points and the weihts are only studied for p = 1,...,4 since its setup becomes computationally more and more expensive for hiher orders of the Ansatz. The two error curves correspondin to the smart octree method show that the error in enery norm is independent of the order of the quadrature rule employed. The reason for this behavior is that the discretization error dominates the interation error, even for hih values of p. relative error in enery norm [%] 10 1 octree interation moment fittin - APD moment fittin - GLP moment fittin - OP smart octree - p q =p smart octree - p q =p total number of derees of freedom [-] Fi. 23: Relative error in enery norm Fi. 24 shows the total number of interation points n. The fiure reveals that we can reduce the number of interation points sinificantly usin the moment fittin and, thus, obtain more efficient quadrature rules as compared to the standard interation in the FCM based on an octree subdivision. Applyin the optimization of points and weihts, we obtain even more efficient quadrature rules, althouh the ain is not that hih anymore. Tab. 4 lists the total number of the points of the individual interation methods. Here, p denotes the order of the FCM analysis, n OT the number of points of the octree- and smart octree subdivision (with x denot- and n SOT[x]