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The search for multivariate quadrature rules of minimal size with a specified polynomial accuracy has been the topic of many years of research. Finding such a rule allows accurate integration of moments, which play a central role in many aspects of scientific computing with complex models. The contribution of this paper is twofold. First, we provide novel mathematical analysis of the polynomial quadrature problem that provides a lower bound for the minimal possible number of nodes in a polynomial rule with specified accuracy. We give concrete but simplistic multivariate examples where a minimal quadrature rule can be designed that achieves this lower bound, along with situations that showcase when it is not possible to achieve this lower bound. Our second contribution is the formulation of an algorithm that is able to efficiently generate multivariate quadrature rules with positive weights on non-tensorial domains. Our tests show success of this procedure in up to 20 dimensions. We test our method on applications to dimension reduction and chemical kinetics problems, including comparisons against popular alternatives such as sparse grids, Monte Carlo and quasi Monte Carlo sequences, and Stroud rules. The quadrature rules computed in this paper outperform these alternatives in almost all scenarios. [ABSTRACT FROM AUTHOR]

Meshfree approximation schemes possess a high potential in computer aided engineering due to their large flexibility. Especially the tremendous progress in processor technology within recent years relativizes the increase in computation time due to the inherent search algorithm. Nevertheless meshfree approximation schemes are still faced with some challenges, like imposition of Dirichlet boundary conditions, robustness of the algorithm and accuracy. The recent developed Optimal Transportation Meshfree (OTM) method seemed to overcome most of these problems. In this paper the OTM solution scheme is combined with a standard search algorithm in order to allow a simple and flexible computation. However this scheme is not stable for some examples of application. Hence an investigation is conducted which shows that the reason for this instability is due to underintegration. Based on this investigation a remedy to stabilize the algorithm is suggested which is based on well known concepts to control the hourglass effects in the Finite Element Method. In contrast to the original publication, the OTM algorithm is derived here from the principle of virtual work. Local maximum entropy shape functions are used which possess a weak Kronecker- δ property. This enables a direct imposition of Dirichlet boundary conditions if the boundary is convex. The limitations of this basis function are also addressed in this paper. Additionally, the search algorithm presented fulfills basic topological requirements. Several examples are investigated demonstrating the improved behavior of the stabilized OTM algorithm. [ABSTRACT FROM AUTHOR]

The present paper focuses on the discretization and implementation of the Thick Level Set (TLS) approach for quasi-brittle failure when dealing with both local and non-local development of damage. The TLS damage model introduces a length scale and nonlocality in the Fracture Process Zone (FPZ) and it handles the continuous to discontinuous failure transition. The added nonlocality is limited to the FPZ and therefore the model is local away from the FPZ or before its emergence. It leads to concurrent development of local and nonlocal damage during simulations. This paper presents a space–time discretization that handles concurrent computations in a unified manner. It is based on an explicit time discretization written as a predictor–corrector scheme and a space discretization that uses specific approximation functions (modes) that embed nonlocality. It also uses an alternative implementation of the eXtended Finite Element Method (X-FEM) enrichment used in the TLS which is based on virtual nodes. [ABSTRACT FROM AUTHOR]

This paper presents an effective method to automatically construct trivariate tensor-product spline models of complicated geometry and arbitrary topology. Our method takes as input a solid model defined by its triangulated boundary surface. Using cuboid decomposition, an initial polycube approximating the input boundary mesh is built. This polycube serves as the parametric domain of the tensor-product spline representation required for isogeometric analysis. The polycube’s nodes and arcs decompose the input model’s boundary into quadrilateral patches, and these patches form hexahedral domains. Using aligned global parameterization, the nodes are re-positioned and the arcs are re-routed across the surface in a way to achieve low overall patch distortion, and alignment to principal curvature directions and sharp features. The optimization process is based on one of the main contributions of this paper: a novel way to design cross fields with topological (i.e., imposed singularities) and geometrical (i.e., imposed directions) constraints by solving only sparse linear systems. Based on the optimized polycube and parameterization, compatible B-spline boundary surfaces are reconstructed. Finally, the interior volumetric parameterization is computed using Coon’s interpolation. In the context of parametric studies based on geometrical parameters, this method can be used to compute the morphing required for reduced order modeling. For different parametric instances with the same topology but different geometries, this method allows to have the same representation: i.e., meshes (or parameterizations) with the same topology. The efficiency and the robustness of the proposed approach are illustrated by several examples. [ABSTRACT FROM AUTHOR]

In the present paper, the Stefan problem is solved by enriching an underlying NURBS-based isogeometric approximation with an explicitly defined (sharp) interface on which a hybrid function/derivative condition is isoparametrically described. Since the geometry of the enrichment is explicitly defined, normals and curvatures are explicitly computed at any point on the interface. Thus, the enriched approximation naturally captures the interfacial discontinuity in temperature gradient and naturally enables the imposition of Gibbs–Thomson condition. The blending of the enrichment with the underlying approximation requires an estimate of distance to the enriching geometry from a quadrature point and the parametric value of the footpoint on the enriching geometry. These quantities are computed efficiently in the present paper using an algebraic estimate of distance coupled with an algebraic point projection method. These algebraic schemes rely on implicitization of the parametric curve, and are shown to be more efficient and robust than Newton–Raphson iterations. Procedures for adaptive time stepping, refinement and coarsening of geometry are developed to increase the stability and efficiency of the developed methodology. Several numerical examples of classical and dendritic Stefan problem are presented to demonstrate the methodology. [ABSTRACT FROM AUTHOR]

This paper presents a hybrid finite-element/boundary-element method for fluid–structure-interaction simulations of inflatable structures. The flow model consists of the steady Stokes equation, which admits a boundary-integral formulation. The structure is represented by a Kirchhoff–Love shell. The boundary-element approximation of the Stokes equation reduces the flow problem to an integral equation on the actual structure configuration, thus obviating the need for volumetric meshing of the strongly deforming fluid domain. The Stokes model moreover exhibits a lubrication effect that acts as an intrinsic mechanism to treat the ubiquitous self-contact that occurs in inflation problems. The aggregated fluid–structure-interaction problem, composed of the boundary-integral equation and the Kirchhoff–Love shell connected by dynamic and kinematic interface conditions, is approximated by means of isogeometric discretizations to accommodate the smoothness requirements on the approximation spaces imposed by the flexural rigidity in the Kirchhoff–Love shell and to provide an accurate and smooth representation of the boundary for the boundary-element method. Auxiliary results presented in this paper are: (1) a parametrization-free Kirchhoff–Love formulation; (2) establishment of a cubic relationship between distance and tractions due to the lubrication effect; and (3) the interpretation of the Lagrange multiplier pertaining to fluid incompressibility as the total excess pressure. [ABSTRACT FROM AUTHOR]

In this paper we couple collocated Boundary Element Methods (BEM) with unstructured analysis-suitable T-spline surfaces for solving a linear Boundary Integral Equation (BIE) arising in the context of a ship-hydrodynamic problem, namely the so-called Neumann–Kelvin problem, following the formulation by Brard (1972) and Baar and Price (1988). The local-refinement capabilities of the adopted T-spline bases, which are used for representing both the geometry of the hull and approximating the solution of the associated BIE, in accordance with the Isogeometric concept proposed by Hughes et al. (2005), lead to a solver that achieves the same error level for many fewer degrees of freedom as compared with the corresponding NURBS-based Isogeometric-BEM solver recently developed in Belibassakis et al. (2013). In this connection, this paper makes a step towards integrating modern CAD representations for ship-hulls with hydrodynamic solvers of improved accuracy and efficiency, which is a prerequisite for building efficient ship-hull optimizers. [ABSTRACT FROM AUTHOR]

Abstract: This is the second paper out of a series of papers devoted to model- and hpq-adaptive finite element methods assigned for the modeling and analysis of elastic structures of complex mechanical description. In our previous publication we investigated the issue of hierarchical models and approximations of such structures. We applied 3D or 3D-based mechanical models, hierarchical modeling, and hierarchical approximations within the proposed finite element formulation. Furthermore, we assumed that the mechanical model and discretization parameters (such as: the size h of the element, and the longitudinal and transverse approximation orders, p and q) could vary locally, i.e. they could be different in each finite element. The a posteriori error estimation discussed in the present paper is based on the generalization of the residual equilibration method on the models with internal constraints. The generalized method is applied to the assessment of the total and approximation errors, while the modeling error is calculated as the difference between the former two errors. The corresponding error-controlled adaptive procedures are based on a three-step strategy, with possible iterations of h- and p-steps. [Copyright &y& Elsevier]

Abstract: A known problem of partition of unity-based generalized finite element methods is the linear dependence of the approximation space, which leads to singular stiffness matrix. Up to now, the linear dependence problem has not been fully understood and an efficient way to alleviate it is not available. In our previous paper “Prediction of rank deficiency in partition of unity-based methods with plane triangular or quadrilateral meshes” [Comput. Methods Appl. Mech. Engrg. 200 (2011) 665–674], the origin of the linear dependence problem was first dissected and then a method was proposed to reliably predict the rank deficiency of the linearly dependent global approximations of two-dimensional partition of unity-based generalized finite element methods. This paper extends the previous work to three-dimensional cases. The linear dependence problem is first investigated at an element level and then extended to the whole mesh. Derivation of general formulations in a three-dimensional setting is undoubtedly more challenging than a two-dimensional setting because of the complicated element topology. The principle of the increase of rank deficiency is once more applied. The methodology of summing up the added rank deficiency of each element as that of the whole mesh is further proved to be valid in three-dimensional cases. This work together with the previous work is regarded as the essential step to successfully and completely solve the linear dependence problem in partition of unity-based finite element methods. [Copyright &y& Elsevier]

Abstract: This paper establishes the superconvergence and the related recovery type a posteriori error estimators based on projection method for finite element approximation of the elliptic eigenvalue problems. The projection method is a postprocessing procedure that constructs a new approximation by using the least squares method. The results are based on some regularity assumption for the elliptic problem, and are applicable to the finite element approximations of self-adjoint elliptic eigenvalue problems with general quasi-regular partitions. Therefore, the result of this paper can be employed to provide useful a posteriori error estimators in adaptive finite element computation under unstructured meshes. [Copyright &y& Elsevier]

Abstract: This paper considers the formulation of finite elements for thin Kirchhoff rods in the three-dimensional case, extending our work on plane rods presented in . The new issue to be addressed, besides the basic consideration of the three-dimensional geometry, is the approximation of the rod’s torsion, still in the geometrically linear setting of infinitesimal strains. In particular, it is the goal of this paper to identify approximations that preserve the basic invariance properties of the underlying structural theory, in the sense that the fundamental rigid-body modes of translations and infinitesimal rotations are represented exactly without straining, accounting fully for the rod’s curvature and associated Kirchhoff’s kinematics in contrast to common straight (framework) element approximations based on the simplified Euler–Bernoulli beam kinematics. To this purpose, different basic and new interpolations of the angle of twist rotation as well as interpolations of directors increments in the setting provided by a Cosserat treatment of the rod’s kinematics are explored and studied in detail, including rigorous analytical characterizations of the aforementioned rigid-body modes. These different interpolations are combined with the new C 1-continuous Hermitian interpolation developed in that reference for the axis displacements, extended to general three-dimensionally curved rods, as well as with an assumed strain mixed treatment to improve the overall performance of the finite elements. Several representative numerical simulations are presented to illustrate the properties and the numerical performance of the different new finite elements. [Copyright &y& Elsevier]

Abstract: In this paper, we are concerned with meshfree particle methods for the solutions of the classical plate model. The vertical displacement of a thin plate is governed by a fourth order elliptic equation and thus the approximation functions for numerical solutions are required to have continuous partial derivatives. Hence, the conventional finite element method has difficulties to solve the fourth order problems. Meshfree methods have the advantage of constructing smooth approximation functions, however, most of the earlier works on meshfree methods for plate problems used either moving least squares method with penalty method or coupling FEM with meshfree method to deal with essential boundary conditions. In this paper, by using generalized product partition of unity, introduced by Oh et al., we introduce meshfree particle methods in which approximation functions have high order polynomial reproducing property and the Kronecker delta property. We also prove error estimates for the proposed meshfree methods. Moreover, to demonstrate the effectiveness of our method, results of the proposed method are compared with existing results for various shapes of plates with variety of boundary conditions and loads. [Copyright &y& Elsevier]

Abstract Window functions provide a base for the construction of approximation functions in many meshfree methods. They control the smoothness and extent of the approximation functions and are commonly defined using Euclidean distances which helps eliminate the need for a meshed discretization, simplifying model development for some classes of problems. However, for problems with complicated geometries such as nonconvex or multi-body domains, poor solution accuracy and convergence can occur unless the extents of the window functions, and thus approximation functions, are carefully controlled, often a time consuming or intractable task. In this paper, we present a method to provide more control in window function design, allowing efficient and systematic handling of complex geometries. "Conforming" window functions are constructed using Bernstein–Bézier splines defined on local triangulations with constraints imposed to control smoothness. Graph distances are used in conjunction with Euclidean metrics to provide adequate information for shaping the window functions. The conforming window functions are demonstrated using the Reproducing Kernel Particle Method showing improved accuracy and convergence rates for problems with challenging geometries. Conforming window functions are also demonstrated as a means to simplify the imposition of essential boundary conditions. Highlights • Meshfree approximation functions that conform to domain boundaries are constructed. • Nonconvex geometries are robustly handled with improved accuracy. • Method delivers optimal convergence rates for problems with weak discontinuities. • Admissible spaces created, improving boundary condition imposition. [ABSTRACT FROM AUTHOR]

Abstract This paper presents numerical studies with three classes of quadratic Generalized FEM (GFEM) approximations and shows that all of them lead to errors that are orders of magnitude smaller than the FEM with quarter-point elements, provided that appropriate enrichments are selected. However, all of them lead to severely ill-conditioned systems of equations, with a condition number up to O (h − 10) in the case of the GFEM based on a quadratic partition of unity. Enrichment modifications able to address the ill-conditioning of quadratic GFEM approximations while preserving their optimal convergence are proposed. A robust enrichment modification strategy based on a discontinuous FE interpolant is proposed to control the conditioning of branch function enrichment. The discontinuous FE interpolant is a generalization of the continuous one used with the Stable GFEM (SGFEM). We show that SGFEM spaces based on p -hierarchical FEM enrichments are the same as their GFEM counterparts. This guarantees that both GFEM and SGFEM spaces will lead to the same solution. This is not the case for the other classes of second-order spaces. The robustness of the proposed approximation spaces with respect to the crack location in the mesh is also demonstrated. Highlights • First-order GFEM is not competitive with quarter-point FEMs. • Quadratic GFEMs are more accurate than FEM but can be severely ill-conditioned. • Proposed SGFEM provides same conditioning as FEM and same accuracy as GFEM. • III-conditioning of Heaviside functions can be controlled without a FE interpolant. • SGFEM with discontinuous FE interpolant is robust with respect to crack position in the mesh. [ABSTRACT FROM AUTHOR]

Abstract This paper deals with the a priori error analysis for a convection-dominated diffusion 2D problem, when applying the HDG method on a family of anisotropic triangulations. It is known that in this case, boundary or interior layers may appear. Therefore, it is important to resolve these layers in order to recover, if possible, the expected order of approximation. In this work, we extend the use of HDG method on anisotropic meshes. In this context, when the discrete local spaces are polynomials of degree k ≥ 0 , this approach is able to recover an order of convergence k + 1 2 in L 2 for all the variables, under certain assumptions on the stabilization parameter and family of triangulations. Numerical examples confirm our theoretical results. Highlights • We develop an a priori error analysis for a HDG scheme defined on anisotropic meshes that are made of triangles. • We require that the family of triangulations satisfy the maximum angle condition, which is usual in this case. • We include numerical examples that validate our theoretical results. [ABSTRACT FROM AUTHOR]

Abstract Many physical systems are modeled using partial differential equations (PDEs) with uncertain or random inputs. For such systems, naively propagating a fixed number of samples of the input probability law (or an approximation thereof) through the PDE is often inadequate to accurately quantify the "risk" associated with critical system responses. In this paper, we develop a goal-oriented, adaptive sampling and local reduced basis approximation for PDEs with random inputs. Our method determines a set of samples and an associated (implicit) Voronoi partition of the parameter domain on which we build local reduced basis approximations of the PDE solution. The samples are selected in an adaptive manner using an a posteriori error indicator. A notable advantage of the proposed approach is that the computational cost of the approximation during the adaptive process remains constant. We provide theoretical error bounds for our approximation and numerically demonstrate the performance of our method when compared to widely used adaptive sparse grid techniques. In addition, we tailor our approach to accurately quantify the risk of quantities of interest that depend on the PDE solution. We demonstrate our method on an advection–diffusion example and a Helmholtz example. Highlights • We develop a methodology for adaptively constructing local reduced bases. • We provide rigorous derivations of a posteriori errors for risk measures as QoI. • We introduce an implicit Voronoi-partitioning that underpins the concept of local bases. • We demonstrate the performance of the method using a damped Helmholtz problem and an advection–diffusion problem. • We compare the efficiency and accuracy of our method to local and global sparse grid algorithms. [ABSTRACT FROM AUTHOR]

Abstract: This paper presents a residual-based turbulence model for the incompressible Navier–Stokes equations. The method is derived employing the variational multiscale (VMS) framework. A multiscale decomposition of the continuous solution and a priori unique decomposition of the admissible spaces of functions lead to two coupled nonlinear problems termed as the coarse-scale and the fine-scale sub-problems. The fine-scale velocity field is assumed to be nonlinear and time-dependent and is modeled via the bubble functions approach applied directly to the fine-scale sub-problem. A significant contribution in this paper is a systematic and consistent derivation of the fine-scale variational operator, commonly termed as the stabilization tensor that possesses the right order in the advective and diffusive limits, and variationally projects the fine-scale solution onto the coarse-scale space. A direct treatment of the fine-scale problem via bubble functions offers several fine-scale approximation options with varying degrees of mathematical sophistication that are investigated via benchmark problems. Numerical accuracy of the proposed method is shown on a forced-isotropic turbulence problem, statistically stationary turbulent channel flow problems at Re T =395 and 590, and non-equilibrium turbulent flow around a cylinder at Re =3,900. [Copyright &y& Elsevier]

In this work, we develop and analyse a novel Hybrid High-Order discretisation of the Brinkman problem. The method hinges on hybrid discrete velocity unknowns at faces and elements and on discontinuous pressures. Based on the discrete unknowns, we reconstruct inside each element a Stokes velocity one degree higher than face unknowns, and a Darcy velocity in the Raviart–Thomas–Nédélec space. These reconstructed velocities are respectively used to formulate the discrete versions of the Stokes and Darcy terms in the momentum equation, along with suitably designed penalty contributions. The proposed construction is tailored to yield optimal error estimates that are robust throughout the entire spectrum of local (Stokes- or Darcy-dominated) regimes, as identified by a dimensionless number which can be interpreted as a friction coefficient. The singular limit corresponding to the Darcy equation is also fully supported by the method. Numerical examples corroborate the theoretical results. This paper also contains two contributions whose interest goes beyond the specific method and application treated in this work: an investigation of the dependence of the constant in the second Korn inequality on star-shaped domains and its application to the study of the approximation properties of the strain projector in general Sobolev seminorms. [ABSTRACT FROM AUTHOR]

The efficient solution of coupled PDEs/ODEs problems arising in cardiac electrophysiology is of key importance whenever interested to study the electrical behavior of the tissue for several instances of relevant physical and/or geometrical parameters. This poses significant challenges to reduced order modeling (ROM) techniques –such as the reduced basis method –traditionally employed when dealing with the repeated solution of parameter dependent differential equations. Indeed, the nonlinear nature of the problem, the presence of moving fronts in the solution, and the high sensitivity of this latter to parameter variations, make the application of standard ROM techniques very problematic. In this paper we propose a local ROM built through a k -means clustering in the state space of the snapshots for both the solution and the nonlinear term. Several comparisons among alternative local ROMs on a benchmark test case show the effectivity of the proposed approach. Finally, the application to a parametrized problem set on an idealized left-ventricle geometry shows the capability of the proposed ROM to face complex problems. [ABSTRACT FROM AUTHOR]

The most common discretization method for peridynamic models used in engineering problems is the node-based meshfree approach. This method discretizes peridynamic domains by a set of nodes, each associated with a nodal cell with a characteristic volume, leading to a particle-based description of continuum systems. The behavior of each particle is then considered representative of its cell. This limits the convergence rate to the first order. In this paper, we introduce a reproducing kernel (RK) approximation to the field variables in the peridynamic equations to increase the order of convergence of peridynamic numerical solutions. The numerical results demonstrate improved convergence rates in static peridynamic problems using the proposed method. [ABSTRACT FROM AUTHOR]

Local damage models with softening needs localization limiters to preserve the mathematical and physical consistency. In this paper we compare the properties of strain-gradient and damage-gradient regularizations. Gradient-damage models introduce a quadratic dependency of the dissipated energy on the gradient of the damage field and are nowadays extensively used as phase-field approximation of brittle fracture. Their key feature is to provide a smeared approximation of a crack as a band of localized damage with a finite energy dissipation per unit of surface, that can be identified with the fracture toughness of the Griffith model. Strain gradient models introduce a quadratic dependence of the elastic energy on the gradient of the strain field. A similar term can be physically interpreted as the presence in the material of linear, but nonlocal, stiffnesses, that can be eventually be affected by damage. Despite this attractive interpretation, we have found that strain-gradient regularized models can hardly be used to approximate brittle fracture, because smeared cracks with non-vanishing and finite dissipated energies are hardly obtained. Our analysis is based on variational models and focuses on the one-dimensional traction problem. [ABSTRACT FROM AUTHOR]

Abstract: This is a paper out of a series of three papers devoted to model- and hpq-adaptive finite element methods assigned for modeling and analysis of complex structures. In this paper we focus our attention on hierarchical models and hierarchical approximations, while the issues of error estimation and adaptivity control will be presented in the next papers of the series. In all the papers we consider the elastic structures of complex mechanical description. We apply 3D or 3D-based mechanical models, hierarchical modeling, and hierarchical approximations within our finite element formulation. In the proposed approach, the mechanical model and discretization parameters, such as the size h of the element, and the longitudinal and transverse approximation orders, p and q, can vary locally, i.e. they can be different in each finite element. The a posteriori error estimation is based on the residual equilibration methods applied to the assessment of the modeling and approximation errors. The error-controlled adaptive procedures are derived from a four-step strategy, with possible iterations on h- and p-steps. [ABSTRACT FROM AUTHOR]

Abstract: This paper discusses stabilization of an optimization procedure to determine an appropriate semivariogram parameter using the empirical semivariogram for Kriging-based approximation. From a viewpoint of a computational cost for constructing a surrogate model, a semivariogram fitting approach using the empirical semivariogram can be usable for the parameter estimation of a semivariogram function. However, instability of the optimization procedure for determination of the semivariogram parameter may be observed in some cases and it causes to generate an invalid surrogate model. For this problem, a simple technique for stabilization of the optimization for the parameter determination is proposed in this paper. The proposed approach employs a normalization technique of input data with respect to values of variables and an objective function. The proposed method is applied to some numerical examples, and the numerical results illustrate validity and effectiveness of the proposed method. [Copyright &y& Elsevier]

Abstract: In this paper, heat transfer problems exhibiting sharp thermal gradients are analyzed using the classical and generalized finite element methods. The effect of solution roughness on the ability of the methods to obtain accurate approximations is investigated. Convergence studies show that low order (linear and quadratic) elements require strongly refined meshes for acceptable accuracy. We propose a generalized FEM with global–local enrichments for the class of problems investigated in the paper. In this procedure, a global solution space defined on a coarse mesh is enriched through the partition of unity framework of the generalized FEM with solutions of local boundary value problems. The local problems are defined using the same procedure as in the global–local FEM, where boundary conditions are provided by a coarse scale global solution. Coarse, uniform, global meshes are acceptable even at regions with thermal spikes that are orders of magnitude smaller than the element size. Convergence on these discretizations was achieved even when no or limited convergence was observed in the local problems. Two approaches are proposed to improve the boundary conditions prescribed on local problems and their convergence. The use of the corresponding improved local solutions as enrichments for the global problem extends the range of target error level for the enriched global problem. The two-way information transfer provided by the proposed generalized FEM is appealing to several classes of problems, especially those involving multiple spatial scales. The proposed methodology brings the benefits of generalized FEM to problems where limited or no information about the solution is known a priori. [Copyright &y& Elsevier]

Abstract: We begin the mathematical study of the k-method utilizing the theory of Kolmogorov n-widths. The k-method is a finite element technique where spline basis functions of higher-order continuity are employed. It is a fundamental feature of the new field of isogeometric analysis. In previous works, it has been shown that using the k-method has many advantages over the classical finite element method in application areas such as structural dynamics, wave propagation, and turbulence. The Kolmogorov n-width and sup–inf were introduced as tools to assess the effectiveness of approximating functions. In this paper, we investigate the approximation properties of the k-method with these tools. Following a review of theoretical results, we conduct a numerical study in which we compute the n-width and sup–inf for a number of one-dimensional cases. This study sheds further light on the approximation properties of the k-method. We finish this paper with a comparison study of the k-method and the classical finite element method and an analysis of the robustness of polynomial approximation. [Copyright &y& Elsevier]

Abstract: In this paper, a finite-difference scheme for incompressible flow problems is treated. The scheme uses non-staggered grid for velocity approximation. A special stabilization is introduced to ensure the well-posedness and optimal approximation properties of the scheme. The stability estimate is proved in the form of a mesh-independent bound for the norm of discrete operator inverse. The finite-difference method is particularly suitable for problems in which velocity vector is involved in additional quantities that enter the system of flow equations as, for example, in the Bingham problem. We describe this application in the paper in some detail. Results of numerical experiments are included that confirm the applicability and optimality of the method. [Copyright &y& Elsevier]

Abstract: A spatial stabilization via bubble functions of linear finite element methods for nonlinear evolutionary convection–diffusion equations is discussed. The method of lines with SUPG discretization in space leads to numerical schemes that are not only difficult to implement, when considering nonlinear evolutionary equations, but also do not produce satisfactory results. The method we propose can be seen as an alternative to this kind of methods. Once the numerical approximation belonging to a linear finite element space enriched with bubble functions is obtained, the bubble part is discarded. The linear part is shown to give a stabilized approximation to the solution being approached. The bubble functions are deduced using a linear steady convection–diffusion model in such a way that the linear part of the approximation to the linear steady model (after static condensation of bubbles) reproduces the SUPG method in the convection-dominated regime. However, for the nonlinear evolutionary equations we consider in the paper the method we propose is not equivalent to the SUPG method. Some numerical experiments are provided in the paper to show the efficiency of the procedure. [Copyright &y& Elsevier]

Abstract: This paper considers the spectral element approximation of the Stokes problem and the conditioning of the resulting discrete problem. The well-posedness of the variational formulation of the Stokes problem and, in particular, the uniqueness of the pressure has been demonstrated when the subspace of square-integrable functions having vanishing mean is chosen for the pressure space. In the discrete setting a conforming subspace is chosen for the discrete pressure space and the analysis of the discrete problem is based on this choice. However, this space is not used in practical implementations of finite or spectral element methods for the Stokes problem. In this paper it is shown how the zero volume condition on the pressure may be accommodated within the trial space by modifying the continuity equation in a consistent manner. The dependence of the condition number of the Uzawa operator and the accuracy of the spectral element approximation on the weighting factor is investigated. [Copyright &y& Elsevier]

This paper presents a new isogeometric topology optimization (TO) method based on moving morphable components (MMC), where the R-functions are used to represent the topology description functions (TDF) to overcome the C 1 discontinuity problem of the overlapping regions of components. Three new ersatz material models based on uniform, Gauss and Greville abscissae collocation schemes are used to represent both the Young’s modulus of material and the density field based on the Heaviside values of collocation points. Three benchmark examples are tested to evaluate the proposed method, where the collocation schemes are compared as well as the difference between isogeometric analysis (IGA) and finite element method (FEM). The results show that the convergence rate using R-functions has been improved in a range of 17%–60% for different cases in both FEM and IGA frameworks, and the Greville collocation scheme outperforms the other two schemes in the MMC-based TO. [ABSTRACT FROM AUTHOR]

This paper marks the debut of a Galerkin isogeometric method for solving a Fredholm integral eigenvalue problem, enabling random field discretization by means of the Karhunen–Loève expansion. The method involves a Galerkin projection onto a finite-dimensional subspace of a Hilbert space, basis splines (B-splines) and non-uniform rational B-splines (NURBS) spanning the subspace, and standard methods of eigensolutions. Compared with the existing Galerkin methods, such as the finite-element and mesh-free methods, the NURBS-based isogeometric method upholds exact geometrical representation of the physical or computational domain and exploits regularity of basis functions delivering globally smooth eigensolutions. Therefore, the introduction of the isogeometric method for random field discretization is not only new; it also offers a few computational advantages over existing methods. In the big picture, the use of NURBS for random field discretization enriches the isogeometric paradigm. As a result, an uncertainty quantification pipeline of the future can be envisioned where geometric modeling, stress analysis, and stochastic simulation are all integrated using the same building blocks of NURBS. Three numerical examples, including a three-dimensional random field discretization problem, illustrate the accuracy and convergence properties of the isogeometric method for obtaining eigensolutions. [ABSTRACT FROM AUTHOR]

A stable basis for isogeometric analysis of trimmed models is obtained by combining extended B-splines with truncated hierarchical B-splines. While extended B-splines guarantee that the condition number of system matrices is independent of the location of a trimming curve, local refinement is used to improve the robustness of the procedure and the accuracy of the numerical results. The present extended B-spline construction works in the context of Galerkin and collocation methods. The paper focuses on the latter and introduces a new collocation scheme for truncated hierarchical B-splines. A proper transition between refinement levels is assured by a novel balancing algorithm that employs a simple criterion. The enhanced performance of the locally refined stabilization is verified by scalar Laplace and linear elasticity problems analyzed by a collocation based isogeometric boundary element method. The proposed approach yields excellent results and requires few refinement levels to improve the stabilization procedure and accuracy along trimming curves. [ABSTRACT FROM AUTHOR]

In this paper, we define analysis-suitable++ (AS++) T-splines which include analysis-suitable (AS) T-splines as a special case and maintain all the good mathematical properties as AS T-splines. We prove that AS++ T-splines are always linear independent regardless of the knot values and show that the classical construction of the dual basis for tensor-product B-splines and AS T-splines can be generalized to AS++ T-spline spaces. We also discuss how all of these issues pave the way to a mathematical theory for AS++ T-splines. [ABSTRACT FROM AUTHOR]

The optimal designs obtained from the deterministic topology optimization without considering the loading uncertainties may become vulnerable, or even lead to catastrophic failures. A two-level optimization formulation is often used in the Robust Topology Optimization (RTO) under uncertain loads. Various approaches have been reported to identify the critical loads associated with the worst structure responses. Because Convex Model approaches apply convex approximations to the original non-convex model at the lower level, the optimal designs obtained by these methods are greatly dependent on the quality of the approximation. In this paper, a new formulation based on the Wolfe duality for the RTO problems with multiple independent unknown-but-bounded loads is proposed. Following the two-level formulation, the lower level optimization problem for the worst multiple independent uncertain loading case is transformed by the Wolfe duality. Both the first order necessary conditions and the second order sufficient conditions are derived rigorously to validate the solution optimality despite of the non-convexity associated with the lower level formulation. Numerical examples are also presented to demonstrate the proposed approach. [ABSTRACT FROM AUTHOR]

The main aim of the current paper is to propose a new truly meshless numerical technique to solve the one- and two-dimensional elliptic interface problems. The employed meshless approach is based on a new class of MLS approximation. The numerical procedure is based on the interpolating stabilized MLS (ISMLS) approximation. The new shape functions that have been made by the ISMLS technique, have the δ -Kronecker property thus the Dirichlet boundary conditions can be applied, directly. In the current investigation, we propose a meshless collocation method for solving elliptic interface problems based on the shape functions of ISMLS technique. Several test problems with sufficient complexity have been studied to check the accuracy and efficiency of the new numerical procedure. Moreover, test problems show the acceptable accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

We define an a posteriori verification procedure that enables to control and certify PGD-based model reduction techniques applied to parametrized linear elliptic or parabolic problems. Using the concept of constitutive relation error, it provides guaranteed and fully computable global/goal-oriented error estimates taking both discretization and PGD truncation errors into account. Splitting the error sources, it also leads to a natural greedy adaptive strategy which can be driven in order to optimize the accuracy of PGD approximations. The focus of the paper is on two technical points: (i) construction of equilibrated fields required to compute guaranteed error bounds; (ii) error splitting and adaptive process when performing PGD-based model reduction. Performances of the proposed verification and adaptation tools are shown on several multi-parameter mechanical problems. [ABSTRACT FROM AUTHOR]

When relying on Newton iterations to solve nonlinear problems in the context of Reduced Basis (RB) methods, the assembling of the RB arrays during the online stage depends on the dimension of the underlying high-fidelity approximation. This is more of an issue when dealing with fully nonlinear problems, for which the global Jacobian matrix has to be entirely reassembled at each Newton step. In this paper, the Discrete Empirical Interpolation Method (DEIM) and its matrix version (MDEIM) are combined to evaluate both the residual vector and the Jacobian matrix very efficiently in the case of complex parametrized nonlinear mechanical problems. We compare this strategy with the classical DEIM approach and we derive a posteriori error estimates on the solution accounting for the contribution of DEIM/MDEIM errors. The effectiveness of the proposed framework is assessed on quasi-static nonlinear problems. In particular, we consider a nonlinear elasticity problem defined on a cube and a mechanical model describing heart contraction, for an idealized left ventricle geometry. The latter is a coupled problem, in which the activation of the heart contraction is given by the solution of an electrophysiology model. Our numerical results show that MDEIM is preferable to the classical DEIM, both in terms of efficiency and accuracy. [ABSTRACT FROM AUTHOR]

This paper presents a displacement-based and a mixed isogeometric collocation formulations for three-dimensional geometrically exact shear-deformable beams with arbitrarily curved initial geometry. Strong forms of both formulations are consistently linearized with respect to the underlying configuration manifold. High efficiency is achieved by using a geometrically consistent kinematic model in which incremental rotations are parametrized through the spatial rotation vector. Several numerical tests demonstrate that the proposed mixed formulation is totally locking-free for any choice of approximation degree. Mixed formulation also exhibits higher robustness in the incremental-iterative algorithm. Elimination of shear locking and low computational cost assured by the combination of isogeometric collocation with a consistent kinematic model result in a very efficient, accurate and geometrically powerful method able to handle geometrically complex beam problems. [ABSTRACT FROM AUTHOR]

Discontinuous Galerkin with finite difference rules (DGFD) is applied to mechanical plane stress state problem. The considered domain is discretized by polygonal mesh. The polygonal elements can be for example a hexagon, pentagon or just quadrangle or triangle. They do not have to be convex and a fish mesh, where the elements have fish shapes, is used. When the elements are rectangular then the orthogonality of Chebyshev basis functions can be utilized. In such a case very high-order approximate solution can be obtained. In this work the approximation order exceeds 10 and reaches 60, which in the latter case means 3600 numbers of degrees of freedom in a single element. The paper is illustrated by a benchmark example in which the exact solution is recovered by DGFD method for various meshes. In the other example the stress concentration is easily recovered by very high-order version of DGFD method. [ABSTRACT FROM AUTHOR]

Current material point method (MPM) particle updates use a PIC approach, a FLIP approach, or a linear combination of PIC and FLIP. A PIC update filters velocity in each time step, which causes unwanted numerical diffusion, while FLIP eliminates that diffusion, but may retain too much noise. This paper develops a new particle update termed XPIC( m ) (for eXtended PIC of order m ) because it generalizes PIC updates. XPIC(1) is identical to current PIC methods, but higher orders of XPIC( m ) address the over filtering and numerical diffusion of PIC, while still filtering out noise caused by the nontrivial null space of the extrapolation matrix used in MPM. As m → ∞ , XPIC( m ) converges to a modified FLIP update with orthogonal removal of null space noise. The frequency response and filtering properties of XPIC( m ) are investigated and several numerical examples demonstrate its advantages over other update methods. [ABSTRACT FROM AUTHOR]

In this paper we introduce, analyze, and compare several approaches designed to incorporate a linear (or affine) constraint within the Proper Generalized Decomposition framework. We apply the considered methods and numerical strategies to two classes of problems: the pure Neumann case where the role of the constraint is to recover unicity of the solution; and the Robin case, where the constraint forces the solution to move away from the already existing unique global minimizer of the energy functional. [ABSTRACT FROM AUTHOR]

In this paper we propose an algorithm for the formation of matrices of isogeometric Galerkin methods. The algorithm is based on three ideas. The first is that we perform the external loop over the rows of the matrix. The second is that we calculate the row entries by weighted quadrature. The third is that we exploit the (local) tensor product structure of the basis functions. While all ingredients have a fundamental role for computational efficiency, the major conceptual change of paradigm with respect to the standard implementation is the idea of using weighted quadrature: the test function is incorporated in the integration weight while the trial function, the geometry parametrization and the PDEs coefficients form the integrand function. This approach is very effective in reducing the computational cost, while maintaining the optimal order of approximation of the method. Analysis of the cost is confirmed by numerical testing, where we show that, for p large enough, the time required by the floating point operations is less than the time spent in unavoidable memory operations (the sparse matrix allocation and memory write). The proposed algorithm allows significant time saving when assembling isogeometric Galerkin matrices for all the degrees of the test spline space and paves the way for a use of high-degree k -refinement in isogeometric analysis. [ABSTRACT FROM AUTHOR]

T-splines developed on unstructured meshes are called unstructured T-splines, which have the problem of geometric inconsistency and unfairness of mesh density evolution over regular or irregular elements during the refinement. Geometric inconsistency means that the geometry defined by an unstructured T-spline keeps changing throughout the refinement. The unfairness of mesh density evolution pertains to different scalings of regular and irregular elements during the refinement, which drags down the convergence of the approximation result in isogeometric analysis. This paper proposes rectified unstructured T-splines to solve these two problems with the dynamic weighted refinement and rectified T-spline basis functions. Owing to the dynamic weighted refinement, the mesh size scaling tends to 1 2 fairly everywhere on unstructured meshes, and geometric consistency is improved with the rectified T-spline basis functions. The approximation convergence is also greatly enhanced for isogeometric analysis from the original unstructured T-splines, which is quite meaningful for engineering simulations. [ABSTRACT FROM AUTHOR]

The proper orthogonal decomposition reduced-order model (POD-ROM) has been widely used as a computationally efficient surrogate model in large-scale numerical simulations of complex systems. However, when it is applied to a Hamiltonian system, a naive application of the POD method can destroy the Hamiltonian structure in the reduced-order model. In this paper, we develop a new reduced-order modeling approach for Hamiltonian systems, which modifies the Galerkin projection-based POD-ROM so that the appropriate Hamiltonian structure is preserved. Since the POD truncation can degrade the approximation of the Hamiltonian function, we propose to use a POD basis from shifted snapshots to improve the approximation to the Hamiltonian function. We further derive a rigorous a priori error estimate for the structure-preserving ROM and demonstrate its effectiveness in several numerical examples. This approach can be readily extended to dissipative Hamiltonian systems, port-Hamiltonian systems, etc. [ABSTRACT FROM AUTHOR]

This paper discusses design-under-uncertainty problems that employ a probabilistic performance as objective function and consider its estimation through stochastic (i.e., Monte Carlo) simulation. This setting facilitates a great modeling flexibility but has an associated high computational burden, and a framework relying on kriging surrogate modeling is proposed here to address this challenge. The metamodel is formulated to approximate the system response (metamodel output) with respect to both the design variables and the uncertain model parameters (augmented metamodel input), so that it can simultaneously support the uncertainty propagation and the design optimization, adopting a sequential approximate optimization (SAO) strategy for the latter. At each SAO cycle a sub-region of the design space is defined and a kriging metamodel is built exploiting information from the previous SAO cycle to improve accuracy. Within a stochastic simulation setting this metamodel is first utilized to approximate the system performance when estimating the objective function for specific design configurations. This information is subsequently used to search for a local optimum within the considered design sub-region till convergence to an interior point or the boundary of the region is established, prompting the termination of the current SAO cycle. When convergence to the boundary is established, the quality of the obtained solution is examined and if desired criteria are satisfied the next cycle moves to a new sub-region and the process is repeated till convergence is established. Else, an adaptive refinement of the design of experiments is established by adding optimally selected new support points for informing the metamodel development, and the next SAO cycle remains in the current sub-region. Through this approach an adaptive control for the accuracy of the metamodel is achieved, minimizing the number of simulations for the expensive system model. [ABSTRACT FROM AUTHOR]

This paper proposes a large eddy simulation reduced order model (LES-ROM) framework for the numerical simulation of convection-dominated flows. In this LES-ROM framework, the proper orthogonal decomposition (POD) is used to define the ROM basis and a ROM differential filter is used to define the large ROM structures. An approximate deconvolution (AD) approach is used to solve the ROM closure problem and develop a new AD-ROM. This AD-ROM is tested in the numerical simulation of the three-dimensional flow past a circular cylinder at a Reynolds number R e = 1000 . [ABSTRACT FROM AUTHOR]

In Cai and Zhang (2009), we introduced and analyzed an improved Zienkiewicz–Zhu (ZZ) estimator for the conforming linear finite element approximation to elliptic interface problems. The estimator is based on the piecewise “constant” flux recovery in the H ( div ; Ω ) conforming finite element space. This paper extends the results of Cai and Zhang (2009) to diffusion problems with full diffusion tensor and to the flux recovery both in piecewise constant and piecewise linear H ( div ) space. [ABSTRACT FROM AUTHOR]

This paper introduces a new computational methodology for determining a-posteriori multi-objective error estimates for finite-element approximations, and for constructing corresponding (quasi-)optimal adaptive refinements of finite-element spaces. As opposed to the classical goal-oriented approaches, which consider only a single objective functional, the presented methodology applies to general closed convex subsets of the dual space and constructs a worst-case error estimate of the finite-element approximation error. This worst-case multi-objective error estimate conforms to a dual-weighted residual, in which the dual solution is associated with an approximate supporting functional of the objective set at the approximation error. We regard both standard approximation errors and data-incompatibility errors associated with incompatibility of boundary data with the trace of the finite-element space. Numerical experiments are presented to demonstrate the efficacy of applying the proposed worst-case multi-objective error estimate in adaptive refinement procedures. [ABSTRACT FROM AUTHOR]

In this paper, the stabilized finite element approximation of the Stokes eigenvalue problems is considered for both the two-field (displacement–pressure) and the three-field (stress–displacement–pressure) formulations. The method presented is based on a subgrid scale concept, and depends on the approximation of the unresolvable scales of the continuous solution. In general, subgrid scale techniques consist in the addition of a residual based term to the basic Galerkin formulation. The application of a standard residual based stabilization method to a linear eigenvalue problem leads to a quadratic eigenvalue problem in discrete form which is physically inconvenient. As a distinguished feature of the present study, we take the space of the unresolved subscales orthogonal to the finite element space, which promises a remedy to the above mentioned complication. In essence, we put forward that only if the orthogonal projection is used, the residual is simplified and the use of term by term stabilization is allowed. Thus, we do not need to put the whole residual in the formulation, and the linear eigenproblem form is recovered properly. We prove that the method applied is convergent, and present the error estimates for the eigenvalues and the eigenfunctions. We report several numerical tests in order to illustrate that the theoretical results are validated. [ABSTRACT FROM AUTHOR]

Non-conforming approximation methods are becoming increasingly popular because of the potential to apply to multi-material and multi-model analysis for both bounded and unbounded domains. In this paper, we present a least-square approximation based method to solve the one or two dimensional elliptic problems on an unbounded domain. The method gives exponential accuracy and shows superior performance when compared to other numerical methods. Differentiability estimates and the main stability estimate theorem, using a non-conforming spectral element method, are also discussed. The exponential convergence rate of the proposed method is also shown through rigorous error estimate and specific numerical examples. [ABSTRACT FROM AUTHOR]

The paper presents investigations on the performance of the finite element numerical integration algorithm for first order approximations and three processor architectures, popular in scientific computing, classical x86_64 CPU, Intel Xeon Phi and NVIDIA Kepler GPU. We base the discussion on theoretical performance models and our own implementations for which we perform a range of computational experiments. For the latter, we consider a unifying programming model and portable OpenCL implementation for all architectures. Variations of the algorithm due to different problems solved and different element types are investigated and several optimizations aimed at proper optimization and mapping of the algorithm to computer architectures are demonstrated. The experimental results show the varying levels of performance for different architectures, but indicate that the algorithm can be effectively ported to all of them. The conclusions indicate the factors that limit the performance for different problems and types of approximation and the performance ranges that can be expected for FEM numerical integration on different processor architectures. [ABSTRACT FROM AUTHOR]