Cubature Scheme Research Articles

Numerical integration in the virtual element method with the scaled boundary cubature scheme

AbstractThe virtual element method (VEM) is a stabilized Galerkin method on meshes that consist of arbitrary (convex and nonconvex) polygonal and polyhedral elements. A crucial ingredient in the implementation of low‐ and high‐order VEM is the numerical integration of monomials and nonpolynomial functions over such elements. In this article, we apply the recently proposed scaled boundary cubature (SBC) scheme to compute the weak form integrals in various virtual element formulations over polygonal and polyhedral meshes. In doing so, we demonstrate the flexibility of the approach and the accuracy that it delivers on a broad suite of boundary‐value problems in 2D and 3D over polytopes with affine faces as well as on elements with curved boundaries. In addition, the use of the SBC scheme is exemplified in an enriched Poisson formulation of the VEM in which weakly singular functions are required to be integrated. This study establishes the SBC method as a simple, accurate and efficient integration scheme for use in the VEM.

Alternating size field optimizing and parameterization domain CAD model remeshing

Tessellating CAD models into triangular meshes is a long-lasting problem. Size field is widely used to accommodate varieties of requirements in remeshing, and it is usually discretized and optimized on a prescribed background mesh and kept constant in the subsequent remeshing procedure. Instead, we propose optimizing the size field on the current mesh, then using it as guidance to generate the next mesh. This simple strategy eliminates the need of building a proper background mesh and greatly simplifies the size field query. For better quality and convergence, we also propose a geodesic distance based initialization and adaptive re-weighting strategy in size field optimization. Similar to existing methods, we also view the remeshing of a CAD model as the remeshing of its parameterization domain, which guarantees that all the vertices lie exactly on the CAD surfaces and eliminates the need for costly and error-prone projection operations. However, for vertex smoothing which is important for mesh quality, we carefully optimize the vertex's location in the parameterization domain for the optimal Delaunay triangulation condition, along with a high-order cubature scheme for better accuracy. Experiments show that our method is fast, accurate and controllable. Compared with state-of-the-art methods, our approach is fast and usually generates meshes with smaller Hausdorff error, larger minimal angle with a comparable number of triangles.

Comment re: Schemes for cubature over the unit disk found via numerical optimization

A recent paper in this journal contained speculation about the potential for improving some of its novel high-order cubature schemes. Those schemes are for approximating uniformly weighted integrals over the unit disk. The two highest order rules were (i) a degree-65 solution with 733 cubature points and (ii) a degree-77 solution with 1021 cubature points. Both schemes possessed six-fold rotational symmetry and it is reported here that, as speculated, one point can indeed be dropped from each segment so that the end results have just 727 and 1015 cubature points, respectively. The details for these new schemes are now presented along with further discussion of this type of solution and additional characterization of related options of even higher orders.

Averaged cubature schemes on the real positive semiaxis

Stratified cubature rules are proposed to approximate double integrals defined on the real positive semiaxis. In particular, anti-Gauss cubature formulae are introduced and averaged cubature schemes are developed. Some of their appropriate modifications are also studied. Several numerical experiments are given to testify the performance of all the formulae.

Convergence rates of RLT and Lasserre-type hierarchies for the generalized moment problem over the simplex and the sphere

We consider the generalized moment problem (GMP) over the simplex and the sphere. This is a rich setting and it contains NP-hard problems as special cases, like constructing optimal cubature schemes and rational optimization. Using the reformulation-linearization technique (RLT) and Lasserre-type hierarchies, relaxations of the problem are introduced and analyzed. For our analysis we assume throughout the existence of a dual optimal solution as well as strong duality. For the GMP over the simplex we prove a convergence rate of O(1/r) for a linear programming, RLT-type hierarchy, where r is the level of the hierarchy, using a quantitative version of Pólya’s Positivstellensatz. As an extension of a recent result by Fang and Fawzi (Math Program, 2020. https://doi.org/10.1007/s10107-020-01537-7) we prove the Lasserre hierarchy of the GMP (Lasserre in Math Program 112(1):65–92, 2008. https://doi.org/10.1007/s10107-006-0085-1) over the sphere has a convergence rate of O(1/r^2). Moreover, we show the introduced linear RLT-relaxation is a generalization of a hierarchy for minimizing forms of degree d over the simplex, introduced by De Klerk et al. (J Theor Comput Sci 361(2–3):210–225, 2006).

Schemes for cubature over the unit disk found via numerical optimization

Cubature schemes, in this case for uniformly weighted integration over the unit disk, enable exact evaluation of numerical integrals of polynomials but have been explicitly constructed for only low or moderate degrees. In this paper, cubature formulae are discovered for a wider range of degrees by leveraging numerical optimization. These results include a degree-17 cubature scheme with fewer points than existing solutions and up to a degree-77 solution with 1021 cubature points. Optimization heuristics and patterns in the distributions of cubature points are discussed, which serve as vital guides in this work. For example, these heuristics leverage a connection to circle-packing configurations to facilitate the discovery of fully symmetric cubature schemes.

A PERFORMANCE EVALUATION OF TRAPEZOIDAL VARIANTS FOR NUMERICAL CUBATURE

In this work, double integration cubature schemes of Trapezoid type have been focused. Recently, some derivative-based Trapezoid-type schemes have been proposed in literature incorporating derivatives at means of the limits of integration. We carry out the exhaustive performance evaluation of the existing closed Newton-Cotes Trapezoidal (CNCT) double integral scheme with its derivative-based variants in recent literature. The derivative-free and derivative-based rules are discussed in basic forms with local error terms and composite forms with global error terms. The performance of the rules on some double integrals in the form of observed order of accuracy, computational costs and error drops demonstrates the encouraging performance of the derivative-based trapezoidal variants over the derivative-free scheme performing numerical experiments.

Scaled boundary cubature scheme for numerical integration over planar regions with affine and curved boundaries

This paper introduces the scaled boundary cubature (SBC) scheme for accurate and efficient integration of functions over polygons and two-dimensional regions bounded by parametric curves. Over two-dimensional domains, the SBC method reduces integration over a region bounded by m curves to integration over m regions (referred to as curved triangular regions), where each region is bounded by two line segments and a curve. With proper (counterclockwise) orientation of the boundary curves, the scheme is applicable to convex and nonconvex domains. Additionally, for star-convex domains, a tensor-product cubature rule with positive weights and integration points in the interior of the domain is obtained. If the integrand is homogeneous, we show that this new method reduces to the homogeneous numerical integration scheme; however, the SBC scheme is more versatile since it is equally applicable to both homogeneous and non-homogeneous functions. This paper also introduces several methods for smoothing integrands with point singularities and near-singularities. When these methods are used, highly efficient integration of weakly singular functions is realized. The SBC method is applied to a number of benchmark problems, which reveal its broad applicability and superior performance (in terms of time to generate a rule and accuracy per cubature point) when compared to existing methods for integration.

AN EFFICIENT TRAPEZOIDAL SCHEME FOR NUMERICAL CUBATURE WITH HERONIAN MEAN DERIVATIVE

This study focuses on the Heronian mean derivative-based numerical cubature scheme to better evaluate double integrals’ infinite limits. The proposed modifications rely on the Trapezoidal-type quadrature and cubature schemes. The aforementioned proposed scheme is important to numerically evaluate the complex double integrals, where the exact value is not available but the approximate values can only be obtained. With regards to higher precision and order of accuracy, the proposed Heronian derivative-based double integral scheme provides efficient results. The discussed scheme, in basic and composite forms, with local and global error terms is presented with necessary proofs with their performance evaluation against conventional Trapezoid rule through some numerical experiments. The consequently observed error distributions of the aforementioned scheme are found to be lower than the conventional Trapezoidal cubature scheme in composite form

A NEW AND EFFICIENT CENTROIDAL MEAN DERIVATIVE-BASED TRAPEZOIDAL SCHEME FOR NUMERICAL CUBATURE

This research presents a new and efficient Centroidal mean derivative-based numerical cubature scheme which has been proposed for the accurate evaluation of double integrals under finite range. The proposed modification is based on the Trapezoidal-type quadrature and cubature rules. The approximate values can only be obtained for some important applications to evaluate the complex double integrals. Higher precision and order of accuracy could be achieved by the proposed scheme. The schemes, in basic and composite forms, with local and global error terms are presented with necessary supporting arguments with their performance evaluation against conventional Trapezoid rule through some numerical experiments. The simultaneously observed error distributions of the proposed schemes are found to be lower than the conventional Trapezoidal cubature scheme in composite form

CLOSED NEWTON-COTES CUBATURE SCHEMES FOR TRIPLE INTEGRALS WITH ERROR ANALYSIS

Most of the problems in applied sciences in engineering contain integrals, not only in one dimension but also in higher dimensions. The complexity of integrands of functions in one variable or higher variables motivates the quadrature and cubature approximations. Much of the work is focused on the literature on single integral quadrature approximations and double integral cubature schemes. On the other hand, the work on triple integral schemes has been quite rarely focused. In this work, we propose the closed Newton-Cotes-type cubature schemes for triple integrals and discuss consequent error analysis of these schemes in terms of the degree of precision and local error terms for the basic form approximations. The results obtained for the proposed triple integral schemes are in line with the patterns observed in single and double integral schemes. The theorems proved in this work on the local error analysis will be a great aid in extending the work towards global error analysis of the schemes in the future.

ERROR ANALYSIS OF CLOSED NEWTON-COTES CUBATURE SCHEMES FOR DOUBLE INTEGRALS

Numerical integration is one of the fundamental tools of numerical analysis to cope with the complex integrals which cannot be evaluated analytically, and for the cases where the integrand is not mathematically known in closed form. The quadrature rules are used for approximating single integrals, whereas cubature rules are used to evaluate integrals in higher dimensions. In this work, we consider the closed Newton-Cotes cubature schemes for double integrals and discuss consequent error analysis of these schemes in terms of the degree of precision, local error terms for the basic form approximations, composite forms and the global error terms. Besides, the computational cost of the implementation of these schemes is also presented. The theorems proved in this work area pioneering investigation on error analysis of such schemes in the literature.

Rapid evaluation of vignetting-free high-performance systems

System assessment for design often involves averages, such as the standard deviation of the wavefront error, which are estimated by ray tracing through a sample of points within the entrance pupil. General-purpose sampling and weighting schemes are presented, and it is also shown that optical design can benefit from tailored versions of these schemes. It turns out that the type of Gaussian quadrature that has long been recognized for efficiency in this domain requires about 40% to 50% more ray tracing to attain comparable accuracy to generic versions of the new schemes. Even greater efficiency gains can be won, however, by tailoring such sampling schemes to the optical context where azimuthal variation in the wavefront is generally weaker than the radial variation. These new schemes are special cases of what is known in the mathematical world as cubature. Our initial results also led to the consideration of simpler sampling configurations that approximate the newfound cubature schemes. We report on the practical application of a selection of such schemes and make observations that aid in the discovery of cubature schemes relevant to optical design of systems with circular pupils.

Coupling microplane-based damage and continuum plasticity models for analysis of damage-induced anisotropy in plain concrete

A novel plastic-damage constitutive model for plain concrete is developed in this paper. For this purpose, the microplane theory is proposed for overcoming the deficiency of available anisotropic continuum plastic-damage models in reproducing the true anisotropic nature of damage in multidimensional loadings. Based on the microplane theory, the degeneration process in concrete is considered along with plastic deformations. Using the principle of strain energy equivalence, a transformation between the nominal and effective states of material is achieved, that results in a decoupled formulation for damage and plasticity. A yield function with multiple internal variables and a non-associative flow function are employed to describe the plastic behavior of concrete. In order to discriminate between the behavior of concrete in tension and compression, stress tensor is decomposed into tensile and compressive parts, and two sets of microplanes are defined for each material point, leading to corresponding independent anisotropic damage tensors in tension and compression. Several numerical examples are analyzed using the proposed model and robustness and accuracy of the formulation are assessed using the available experimental tests. In addition, superiority of the proposed model over the available isotropic and anisotropic formulations is demonstrated by comparisons between the different approaches. The effects of three different cubature schemes on the response of concrete specimens are also analyzed.

Dimensional hyper-reduction of nonlinear finite element models via empirical cubature

We present a general framework for the dimensional reduction, in terms of number of degrees of freedom as well as number of integration points (“hyper-reduction”), of nonlinear parameterized finite element (FE) models. The reduction process is divided into two sequential stages. The first stage consists in a common Galerkin projection onto a reduced-order space, as well as in the condensation of boundary conditions and external forces. For the second stage (reduction in number of integration points), we present a novel cubature scheme that efficiently determines optimal points and associated positive weights so that the error in integrating reduced internal forces is minimized. The distinguishing features of the proposed method are: (1) The minimization problem is posed in terms of orthogonal basis vector (obtained via a partitioned Singular Value Decomposition) rather that in terms of snapshots of the integrand. (2) The volume of the domain is exactly integrated. (3) The selection algorithm need not solve in all iterations a nonnegative least-squares problem to force the positiveness of the weights. Furthermore, we show that the proposed method converges to the absolute minimum (zero integration error) when the number of selected points is equal to the number of internal force modes included in the objective function. We illustrate this model reduction methodology by two nonlinear, structural examples (quasi-static bending and resonant vibration of elastoplastic composite plates). In both examples, the number of integration points is reduced three order of magnitudes (with respect to FE analyses) without significantly sacrificing accuracy.

Subspace integration with local deformations

Subspace techniques greatly reduce the cost of nonlinear simulation by approximating deformations with a small custom basis. In order to represent the deformations well (in terms of a global metric), the basis functions usually have global support, and cannot capture localized deformations. While reduced-space basis functions can be localized to some extent, capturing truly local deformations would still require a very large number of precomputed basis functions, significantly degrading both precomputation and online performance. We present an efficient approach to handling local deformations that cannot be predicted, most commonly arising from contact and collisions, by augmenting the subspace basis with custom functions derived from analytic solutions to static loading problems. We also present a new cubature scheme designed to facilitate fast computation of the necessary runtime quantities while undergoing a changing basis. Our examples yield a two order of magnitude speedup over full-coordinate simulations, striking a desirable balance between runtime speeds and expressive ability.

Harmonic shells

We propose a procedural method for synthesizing realistic sounds due to nonlinear thin-shell vibrations. We use linear modal analysis to generate a small-deformation displacement basis, then couple the modes together using nonlinear thin-shell forces. To enable audio-rate time-stepping of mode amplitudes with mesh-independent cost, we propose a reduced-order dynamics model based on a thin-shell cubature scheme. Limitations such as mode locking and pitch glide are addressed. To support fast evaluation of mid-frequency mode-based sound radiation for detailed meshes, we propose far-field acoustic transfer maps (FFAT maps) which can be precomputed using state-of-the-art fast Helmholtz multipole methods. Familiar examples are presented including rumbling trash cans and plastic bottles, crashing cymbals, and noisy sheet metal objects, each with increased richness over linear modal sound models.

Optimizing cubature for efficient integration of subspace deformations

We propose an efficient scheme for evaluating nonlinear subspace forces (and Jacobians) associated with subspace deformations. The core problem we address is efficient integration of the subspace force density over the 3D spatial domain. Similar to Gaussian quadrature schemes that efficiently integrate functions that lie in particular polynomial subspaces, we propose cubature schemes (multi-dimensional quadrature) optimized for efficient integration of force densities associated with particular subspace deformations, particular materials, and particular geometric domains. We support generic subspace deformation kinematics, and nonlinear hyperelastic materials. For an r-dimensional deformation subspace with O(r) cubature points, our method is able to evaluate subspace forces at O(r(2)) cost. We also describe composite cubature rules for runtime error estimation. Results are provided for various subspace deformation models, several hyperelastic materials (St.Venant-Kirchhoff, Mooney-Rivlin, Arruda-Boyce), and multimodal (graphics, haptics, sound) applications. We show dramatically better efficiency than traditional Monte Carlo integration. CR CATEGORIES: I.6.8 [Simulation and Modeling]: Types of Simulation-Animation, I.3.5 [Computer Graphics]: Computational Geometry and Object Modeling-Physically based modeling G.1.4 [Mathematics of Computing]: Numerical Analysis-Quadrature and Numerical Differentiation.

A physically motivated sparse cubature scheme with applications to molecular density-functional theory

We present a novel approach for performing multi-dimensional integration of arbitrary functions. The method starts with Smolyak-type sparse grids as cubature formulae on the unit cube and uses a transformation of coordinates based on the conditional distribution method to adapt those formulae to real space. Our method is tested on integrals in one, two, three and six dimensions. The three dimensional integration formulae are used to evaluate atomic interaction energies via the Gordon–Kim model. The six dimensional integration formulae are tested in conjunction with the nonlocal exchange-correlation energy functional proposed by Lee and Parr. This methodology is versatile and powerful; we contemplate application to frozen-density embedding, next-generation molecular-mechanics force fields, ‘kernel-type’ exchange-correlation energy functionals and pair-density functional theory.

On the integral convergence of Kergin interpolation on the disk

In this paper, we discuss the integral convergences of Kergin interpolation and its derivatives for the smooth functions defined on the unit disk. We also propose a new kind of cubature scheme and give the algebraic exactness of this cubature scheme.