Fast simplicial quadrature-based finite element operators using Bernstein polynomials

Division algorithms for Bernstein polynomials

Robust arithmetic for multivariate Bernstein-form polynomials

Residual-based a posteriori error estimates were derived within one unifying framework for lowest-order conforming, nonconforming, and mixed finite element schemes in Carstensen [Numer Math 100:617–637, 2005]. Therein, the key assumption is that the conforming first-order finite element space $$V_h^c$$ annulates the linear and bounded residual l written $$V_h^c \subseteq {\rm ker} \ell$$. That excludes particular nonconforming finite element methods (NCFEMs) on parallelograms in that $$V_h^c \not\subset {\rm ker} \ell$$. The present paper generalises the aforementioned theory to more general situations to deduce new a posteriori error estimates, also for mortar and discontinuous Galerkin methods. The key assumption is the existence of some bounded linear operator $$\Pi: V_h^c \rightarrow V_h^{nc}$$ with some elementary properties. It is conjectured that the more general hypothesis (H1)–(H3) can be established for all known NCFEMs. Applications on various nonstandard finite element schemes for the Laplace, Stokes, and Navier–Lame equations illustrate the presented unifying theory of a posteriori error control for NCFEM.

We combine recently developed finite element algorithms based on Bernstein polynomials [M. Ainsworth, G. Andriamaro, and O. Davydov, SIAM J. Sci. Comput., 33 (2011), pp. 3087--3109; R. C. Kirby, Numer. Math., 117 (2011), pp. 631--652] with the explicit basis construction of the finite element exterior calculus [D. N. Arnold, R. S. Falk, and R. Winther, Comput. Methods Appl. Mech. Engrg., 198 (2009), pp. 1660--1672] to give a family of algorithms for the de Rham complex on simplices that achieves stiffness matrix construction and matrix-free action in optimal complexity. These algorithms are based on realizing the exterior calculus bases as short combinations of Bernstein polynomials. Numerical results confirm optimal scaling of the algorithms and favorable comparison with FEniCS at high polynomial order as well. Additional empirical studies show that very high accuracy is achieved in the mixed discretization of the Poisson equation and the Maxwell eigenvalue problem as the polynomial degree increases.

We present the second-order multidimensional Staggered Grid Hydrodynamics Residual Distribution (SGH RD) scheme for Lagrangian hydrodynamics. The SGH RD scheme is based on the staggered finite element discretizations as in [Dobrev et al., SISC, 2012]. However, the advantage of the residual formulation over classical FEM approaches consists in the natural mass matrix diagonalization which allows one to avoid the solution of the linear system with the global sparse mass matrix while retaining the desired order of accuracy. This is achieved by using Bernstein polynomials as finite element shape functions and coupling the space discretization with the deferred correction type timestepping method. Moreover, it can be shown that for the Lagrangian formulation written in non-conservative form, our residual distribution scheme ensures the exact conservation of the mass, momentum and total energy. In this paper we also discuss construction of numerical viscosity approximations for the SGH RD scheme allowing to reduce the dissipation of the numerical solution. Thanks to the generic formulation of the staggered grid residual distribution scheme, it can be directly applied to both single- and multimaterial and multiphase models. Finally, we demonstrate computational results obtained with the proposed residual distribution scheme for several challenging test problems.

Bernstein–Bézier bases for tetrahedral finite elements

This paper analyzes the rate of convergence of the h-p version of the coupling of the finite element and boundary element method for transmission problems with a linear differential operator with variable coefficients in a bounded polyhedral domain \(\Omega_1\) and with constant coefficients in the exterior domain \(\Omega_2 = {\Bbb R}^3 \backslash \overline{\Omega}_1\). This procedure uses the variational formulation of the differential equation in \(\Omega_1\) and involves integral operators on the interface between \(\Omega_1\) and \(\Omega_2\). The finite elements are used to obtain approximate solutions of the differential equation in \(\Omega_1\) and the boundary elements are used to obtain approximate solutions of the integral equations. For given piecewise analytic data we show that the Galerkin solution of this coupling procedure converges exponentially fast in the energy norm if the h-p version is used both for finite elements and boundary elements.

In this chapter, we advocate application of explicit dynamics (i.e. with explicit time stepping) finite element analysis as the finite element procedure of choice for computational biomechanics of the brain. Our recommendations regarding specific formulations, algorithms, and computing hardware for such analysis include: (1) application of total Lagrangian formulation of continuum mechanics (where the derivatives with respect to the spatial coordinates can be pre-computed); (2) dynamic relaxation for problems (such as image registration) that require steady-state solution; (3) low order (underintegrated eight-noded hexahedra with hourglass control and four-noded tetrahedra with averaged nodal pressure to alleviate volumetric locking) for construction of finite element meshes of the brain; (4) implementation of the finite element algorithms on graphics processing units (GPUs) to facilitate real-time computation (through parallel processing/multithreading) on commodity hardware. We provide examples of verification of the discussed finite element algorithms against the reference solutions obtained using the well-established non-linear static finite element procedures. We discuss time-consuming generation of patient-specific finite element meshes and deterioration of the solution accuracy when the elements undergo distortion induced by large deformations as the limitations of finite element method in the context of surgical simulation.

Finite element modeling of the contact of elements preloaded with a bolt and externally loaded with any force

The application of collocation methods for the numerical solution of singularly perturbed ordinary differential equations is investigated. Collocation at Gauss, Radau and Lobatto points is considered, for both initial and boundary value problems for first order systems with constant coefficients. Particular attention is paid to symmetric schemes for boundary value problems; these problems may have boundary layers at both interval ends. Our analysis shows that certain collocation schemes, in particular those based on Gauss or Lobatto points, do perform very well on such problems, provided that a fine mesh with steps proportional to the layers’ width is used in the layers only, and a coarse mesh, just fine enough to resolve the solution of the reduced problem, is used in between. Ways to construct appropriate layer meshes are proposed. Of all methods considered, the Lobatto schemes appear to be the most promising class of methods, as they essentially retain their usual superconvergence power for the smooth, reduced solution, whereas Gauss–Legendre schemes do not. We also investigate the conditioning of the linear systems of equations arising in the discretization of the boundary value problem. For a row equilibrated version of the discretized system we obtain a pleasantly small bound on the maximum norm condition number, which indicates that these systems can be solved safely by Gaussian elimination with scaled partial pivoting.

Static and dynamic analysis of two-phase peridynamic one-dimensional systems is thoroughly investigated in this paper for various boundary conditions by exact and numerical approaches. Such a nonlocal study concerns the axial or torsional responses of two-phase peridynamic rods or the behavior of a two-phase peridynamic string. For a normalized exponential kernel, the integro-differential equation of the two-phase peridynamic problem can be converted as a linear higher-order differential equation with constant coefficients. Exact static solutions of fixed–fixed and fixed–free two-phase peridynamic axial rod (TPPAR) under uniform distributed loading and concentrated force are analytically derived. It is shown that the pure peridynamic rod problem can be ill-posed due to a conflict between the natural boundary condition and the peridynamic constitutive boundary condition. The two-phase peridynamic problem may be justified from physical arguments, and is also seen as a regularized peridynamic problem. The exact eigen-frequencies of the two-phase peridynamic rods are also analytically obtained for various boundary conditions, including fixed–free, fixed–fixed or free–free boundary conditions. These theoretical approaches are corroborated by numerical investigations based on Finite Difference Method (FDM), Finite Element Method (FEM), exact FEM and variational iterative methods. Two FDM schemes are presented for the statics and the vibration of two-phase peridynamic rods, with convergence orders of 1 and 2, respectively. Moreover, three different FE procedures are proposed as: (1) FEM for direct solving the integro-differential equations of two-phase peridynamic one-dimensional system, (2) FEM for discretizing the pure differential equations, and (3) Exact FEM to obtain both the exact stiffness matrix and exact dynamic stiffness matrix of two-phase peridynamic one-dimensional system, based on Eisenberger’s methodology. The order of convergence for the first and the second proposed FEM is equal to 2 and 4, according to the type of selected finite elements. Moreover, the third proposed FEM based on the exact shape function of the peridynamic problem gives the exact solution from one single FEM element. In addition, the variational iterative method of Stodola–Vianello is successfully applied to the two-phase peridynamic eigenvalue problems.

Introduction. This distinctive paper addresses the local semi-analytical solution to the problem of plate analysis. Isotropic plates featuring the regularity (constancy) of physical and geometric parameters (modulus of elasticity of the plate material, Poisson’s ratio of the plate material, dimensions of the cross section of the plate) along one direction (dimension) are under consideration. This direction is conventionally called the basic direction. Materials and methods. The B-spline wavelet discrete-continual finite element method (DCFEM) is used. The initial operational formulation of the problem was constructed using the theory of distribution and the so-called method of extended domain, proposed by Prof. Alexander B. Zolotov. Results. Some relevant issues of construction of normalized basis functions of the B-spline are considered; the technique of approximation of corresponding vector functions and operators within DCFEM is described. The problem remains continual if analyzed along the basic direction, and its exact analytical solution can be obtained, whereas the finite element approximation is used in combination with a wavelet analysis apparatus in respect of the non-basic direction. As a result, we can obtain a discrete-continual formulation of the problem. Thus, we have a multi-point (in particular, two-point) boundary problem for the first-order system of ordinary differential equations with constant coefficients. A special correct analytical method of solving such problems was developed, described and verified in the numerous papers of the co-authors. In particular, we consider the simplest sample analysis of a plate (rectangular in plan) fixed along the side faces exposed to the influence of the load concentrated in the center of the plate. Conclusions. The solution to the verification problem obtained using the proposed version of wavelet-based DCFEM was in good agreement with the solution obtained using the conventional finite element method (the corresponding solutions were constructed with and without localization; these solutions almost completely coincided, while the advantages of the numerical-analytical approach were quite obvious). It is shown that the use of B-splines of various degrees within wavelet-based DCFEM leads to a significant reduction in the number of unknowns.

In this work, analytical solution of simply supported sandwich arches considering permeation effect of adhesives is presented. The permeation layer is described by the functionally graded material, exponentially graded in the radial direction. The stresses and deformations of each layer are based on the two-dimensional (2D) elasticity theory in the polar coordinate. The governing equations of the arch are solved by the layer-wise method, which turns the differential equations with variable coefficients into constant coefficients. The solution can be obtained efficiently by means of the recursive matrix method, especially for the arch with many layers. The present solution agrees well with the finite element solution with a fine mesh, while the finite element method is time consuming in mesh division and calculation. The one-dimensional (1D) solution based on the Euler–Bernoulli theory is close to the present one; however, the error increases as the arch becomes thick. The effect of permeation layer thickness on the stresses is studied. It is indicated that the stress distributions tend to be smooth along the radial direction as the permeation layer thickness increases.

This paper presents an overview of research and development efforts that are currently being devoted to integrate large deformation finite element formulations with flexible multibody system algorithms. The goal is to develop computer simulation capabilities for the analysis of physics and engineering models with significant details. The successful development of such new and integrated algorithms will also allow modeling and simulation of systems that cannot be solved using existing computer algorithms and codes. One of the main difficulties encountered in this integration process is attributed to the fact that the solution procedures used in finite element codes differ significantly from those used in general-purpose flexible multibody system codes. Finite element methods employ the corotational formulations that are often used with incremental solution procedures. Flexible multibody computer codes, on the other hand, do not, in general, use incremental solution procedures. Three approaches are currently being explored by academic institutions and the software industry. In the first approach, gluing algorithms that aim at performing successful simulations by establishing an interface between existing codes are used. Using different coordinates and synchronizing the time stepping are among several challenging problems that are encountered when gluing algorithms are used. In the second approach, multibody system capabilities are implemented in existing finite element algorithms that are based on large rotation vector formulations. For the most part, corotational formulations and incremental solution procedures are used in this case. In the third approach, a new large deformation finite element formulation that can be successfully implemented in flexible multibody system computer algorithms that employ nonincremental solution procedures is introduced. The approach that is now being developed in several institutions is based on the finite element absolute nodal coordinate formulation. Such a formulation can be systematically implemented in general-purpose flexible multibody system computer algorithms. Nonlinear constraint equations that describe mechanical joints between different bodies can be formulated in terms of the absolute coordinates in a straightforward manner. The coupling between the motion of rigid, flexible, and very flexible bodies can also be accurately described. The successful integration of large deformation finite element and multibody system algorithms will lead to a new generation of computer codes that can be systematical and efficiently used in the analysis of many engineering applications.

Finite-element method is widely used for estimation of the structural behavior at various states. The finite-element (FE) models are not, however, a perfect representation of the real-life structures. In order to find a perfect FE model of the structure as well as monitoring of structural parameter changes, the FE model must be updated to match with real-life model. The curse of incomplete measurement is a major challenge for an updating method, especially in frequency domain. This paper proposes and investigates the replacement of the unmeasured degrees of freedom (DOFs) by experimentally enhanced decomposed transfer function (EDTF), in a FRF-based sensitivity equation to mitigate the adverse effects of data incompleteness. The enhanced transfer function is reached by a truncated set of experimentally measured natural frequencies and damping loss factors of real model. To highlight the merits of the EDTF in mitigating the effect of incomplete measurement, it is compared with the dynamic expansion of measured mode shapes. The results of the numerical investigations show that proposed approach is more efficient than dynamic expansion in the presence of high measurement error, large damages (parameter changes) and fewer measured DOFs. Besides, EDTF eliminates the need for modal data measurement up to a frequency much higher than the highest excitation frequency, which is necessary in dynamic expansion method. In this study, linear least-square method bounded with impartial side constraints is used for solving the sensitivity-based error minimization problem needed for model updating. Furthermore, fifty sets of numerically simulated error-contaminated FRF data are adopted to conduct Monte-Carlo analysis of the robustness of model updating results against error. The mean and coefficient of variation of estimated parameters are utilized for results indication and illustration purpose.