Dual Basis Functions Research Articles

Nonuniform Sampling in Lp-Subspaces Associated with the Multi-Dimensional Special Affine Fourier Transform

In this paper, the sampling and reconstruction problems in function subspaces of Lp(Rn) associated with the multi-dimensional special affine Fourier transform (SAFT) are discussed. First, we give the definition of the multi-dimensional SAFT and study its properties including the Parseval’s relation, the canonical convolution theorems and the chirp-modulation periodicity. Then, a kind of function spaces are defined by the canonical convolution in the multi-dimensional SAFT domain, the existence and the properties of the dual basis functions are demonstrated, and the Lp-stability of the basis functions is established. Finally, based on the nonuniform samples taken on a dense set, we propose an iterative reconstruction algorithm with exponential convergence to recover the signals in a Lp-subspace associated with the multi-dimensional SAFT, and the validity of the algorithm is demonstrated via simulations.

Numerical analysis of a history-dependent mixed hemivariational-variational inequality in contact problems

This paper is devoted to a study of a history-dependent mixed hemivariational-variational inequality arising in contact mechanics. The contact problem concerns the deformation of a viscoelastic body with long memory, subject to a general friction law on one part of the boundary and a frictionless Signorini condition on another part of the boundary. The solution existence and uniqueness of the history-dependent mixed hemivariational-variational inequality based on a Lagrange multiplier approach are proved. Then, a fully discrete scheme is introduced and studied. The trapezoidal rule is used to approximate the integral in the history-dependent operator. For the spatial discretization, the linear finite elements are used to discretize the displacement field, and dual basis functions are used in the approximation of the Lagrange multiplier. Optimal order error estimates are derived for the displacement and the Lagrange multiplier under appropriate solution regularity assumptions. Numerical results are presented to illustrate the theoretical prediction of the convergence orders.

Towards higher-order accurate mass lumping in explicit isogeometric analysis for structural dynamics

We present a mass lumping approach based on an isogeometric Petrov–Galerkin method that preserves higher-order spatial accuracy in explicit dynamics calculations irrespective of the polynomial degree of the spline approximation. To discretize the test function space, our method uses an approximate dual basis, whose functions are smooth, have local support and satisfy approximate bi-orthogonality with respect to a trial space of B-splines. The resulting mass matrix is “close” to the identity matrix. Specifically, a lumped version of this mass matrix preserves all relevant polynomials when utilized in a Galerkin projection. Consequently, the mass matrix can be lumped (via row-sum lumping) without compromising spatial accuracy in explicit dynamics calculations. We also address the imposition of Dirichlet boundary conditions and the preservation of approximate bi-orthogonality under geometric mappings. In addition, we establish a link between the exact dual and approximate dual basis functions via an iterative algorithm that improves the approximate dual basis towards exact bi-orthogonality. We demonstrate the performance of our higher-order accurate mass lumping approach via convergence studies and spectral analyses of discretized beam, plate and shell models.

An efficient algorithm for rigid/deformable contact interaction based on the dual mortar method

In a wide range of practical problems, such as forming operations and impact tests, treating one of the contacting bodies as a rigid obstacle is an excellent approximation to the physical phenomenon. In this work, the well-established dual mortar method is adopted to enforce interface constraints in the finite deformation frictionless contact of deformable bodies and rigid obstacles. The efficiency of the non-linear contact algorithm proposed here is based on two main contributions. Firstly, the weighted gap function is modified such that it retains the signal of the discrete gap function. Within the context of rigid/deformable contact, this unlocks a significant simplification by removing the need to explicitly evaluate the dual basis functions. The corresponding first-order interpolation is presented in detail. Particular focus is, then, placed on the extension for second-order interpolation by employing a piecewise linear interpolation scheme, which critically retains the geometrical information of the finite element mesh. Secondly, a new definition for the nodal orthonormal moving frame attached to each contact node is suggested. It reduces the geometrical coupling between the nodes and consequently decreases the stiffness matrix bandwidth. The proposed contributions decrease the computational complexity of dual mortar methods for rigid/deformable interaction, especially in the three-dimensional setting, while preserving accuracy and robustness.

Isogeometric Bézier dual mortaring: The Kirchhoff–Love shell problem

In this paper we develop an isogeometric Bézier dual mortar method for coupling multi-patch Kirchhoff–Love shell structures. The proposed approach weakly enforces the continuity of the solution at patch interfaces through a dual mortar method and can be applied to both conforming and non-conforming discretizations. As the employed dual basis functions have local supports and satisfy the biorthogonality property, the resulting stiffness matrix is sparse. In addition, the coupling accuracy is optimal because the dual basis possesses the polynomial reproduction property. We also formulate the continuity constraints through the Rodrigues’ rotation operator which gives a unified framework for coupling patches that are intersected with G1 continuity as well as patches that meet at a kink. Several linear and nonlinear examples demonstrated the performance and robustness of the proposed coupling techniques.

A hybrid mimetic spectral element method for three-dimensional linear elasticity problems

We introduce a domain decomposition structure-preserving method based on a hybrid mimetic spectral element method for three-dimensional linear elasticity problems in curvilinear conforming structured meshes. The method is an equilibrium method which satisfies pointwise equilibrium of forces. The domain decomposition is established through hybridization which first allows for an inter-element normal stress discontinuity and then enforces the normal stress continuity using a Lagrange multiplier which turns out to be the displacement in the trace space. Dual basis functions are employed to simplify the discretization and to obtain a higher sparsity. Numerical tests supporting the method are presented.

SLIM: A Well-Conditioned Single-Source Boundary Element Method for Modeling Lossy Conductors in Layered Media

The boundary element method (BEM) enables the efficient electromagnetic modelling of lossy conductors with a surface-based discretization. Existing BEM techniques for conductor modelling require either expensive dual basis functions or the use of both single- and double-layer potential operators to obtain a well-conditioned system matrix. The associated computational cost is particularly significant when conductors are embedded in stratified media, and the expensive multilayer Green's function (MGF) must be invoked. In this work, a novel single-source BEM formulation is proposed, which leads to a well-conditioned system matrix without the need for dual basis functions. The proposed single-layer impedance matrix (SLIM) formulation does not require the double-layer potential to model the background medium, which reduces the cost associated with the MGF. The accuracy and efficiency of the proposed method is demonstrated through realistic examples drawn from different applications.

An isogeometric Reissner–Mindlin shell element based on Bézier dual basis functions: Overcoming locking and improved coarse mesh accuracy

We develop a mixed geometrically nonlinear isogeometric Reissner–Mindlin shell element for the analysis of thin-walled structures that leverages Bézier dual basis functions to address both shear and membrane locking and to improve the quality of computed stresses. The accuracy of computed solutions over coarse meshes, that have highly non-interpolatory control meshes, is achieved through the application of a continuous rotational approach. The starting point of the formulation is the modified Hellinger–Reissner variational principle with independent displacement, membrane, and shear strains as the unknown fields. To overcome locking, the strain variables are interpolated with lower-order spline bases while the variations of the strain variables are interpolated with the corresponding Bézier dual bases. Leveraging the orthogonality property of the Bézier dual basis, the strain variables are condensed out of the system with only a slight increase in the bandwidth of the resulting linear system. The condensed approach preserves the accuracy of the non-condensed mixed approach but with fewer degrees of freedom. From a practical point of view, since the Bézier dual basis is completely specified through Bézier extraction, any spline space that admits Bézier extraction can utilize the proposed approach directly.

Isogeometric Bézier dual mortaring: The enriched Bézier dual basis with application to second- and fourth-order problems

In this paper, we present an algorithm to construct enriched Bézier dual basis functions that can reproduce higher-order polynomials. Our construction is unique in that it is based on Bézier extraction and projection, allowing it to be used for tensor product and unstructured polynomial spline spaces, is well-conditioned, and is quadrature-free. When used as a basis for dual mortar methods, optimal approximations are achieved for both second- and fourth-order problems. In the context of fourth-order problems, both C0 and C1 continuity constraints must be applied at each intersection. We develop a novel geometry-independent C1 continuity constraint that also preserves the sparsity of the coupled problem. The performance of the proposed formulation is verified through several challenging second- and fourth-order problems.

Fast Calderón preconditioning for Helmholtz boundary integral equations

Calderón multiplicative preconditioners are an effective way to improve the condition number of first kind boundary integral equations yielding provable mesh independent bounds. However, when discretizing by local low-order basis functions as in standard Galerkin boundary element methods, their computational performance worsens as meshes are refined. This stems from the barycentric mesh refinement used to construct dual basis functions that guarantee the discrete stability of L2-pairings. Based on coarser quadrature rules over dual cells and H-matrix compression, we propose a family of fast preconditioners that significantly reduce assembly and computation times when compared to standard versions of Calderón preconditioning for the three-dimensional Helmholtz weakly and hyper-singular boundary integral operators. Several numerical experiments validate our claims and point towards further enhancements.

A three-field dual mortar method for elastic problems with nonconforming mesh

Conventional two-field mortar methods have master/slave bias and need special treatments for the cross corner problem. In this work, we present a three-field dual mortar method, which employs an intermediate interface to connect individual subdomains with nonconforming meshes. Using this intermediate interface as the master face, a dual mortar method can be formulated by discretizing Lagrange multipliers on the subdomain interfaces using dual basis functions. Furthermore, the computational cost is minimized by static condensation of Lagrange multipliers and elimination of displacement unknowns on subdomain interfaces. Owing to the three-field formulation, the presented method is free of master/slave bias and can handle the cross corner problem naturally. The stability condition for the intermediate interface mesh is investigated by theoretical analysing and numerical modelling. Numerical tests show that the presented method has optimal convergence rate and allows great flexibility in mesh preparation.

Moment Preserving Local Spline Projection Operators

This article describes an elementary construction of a dual basis for nonuniform B-splines that is local, $$L^\infty $$-stable, and reproduces polynomials of any prescribed degree. This allows one to define local projection operators with near-optimal approximation properties in any $$L^q$$, $$1 \le q \le \infty $$, and high order moment preserving properties. As the dual basis functions share the same piecewise polynomial structure as the underlying splines, simple quadrature formulas can be used to compute the projected spline coefficients.

Online basis construction for goal-oriented adaptivity in the generalized multiscale finite element method

In this research, we develop an online enrichment framework for goal-oriented adaptivity within the generalized multiscale finite element method for flow problems in heterogeneous media. The method for approximating the quantity of interest involves construction of residual-based primal and dual basis functions used to enrich the multiscale space at each stage of the adaptive algorithm. Three different online enrichment strategies based on the primal-dual online basis construction are proposed: standard, primal-dual combined and primal-dual product based. Numerical experiments are performed to illustrate the efficiency of the proposed methods for high-contrast heterogeneous problems.

Inverse mass matrix for isogeometric explicit transient analysis via the method of localized Lagrange multipliers

AbstractA variational framework is employed to generate inverse mass matrices for isogeometric analysis (IGA). As different dual bases impact not only accuracy but also computational overhead, several dual bases are extensively investigated. Specifically, locally discontinuous biorthogonal basis functions are evaluated in detail for B‐splines of high continuity and Bézier elements with a standard C0 continuous finite element structure. The boundary conditions are enforced by the method of localized Lagrangian multipliers after generating the inverse mass matrix for completely free body. Thus, unlike inverse mass matrix methods without employing the method of Lagrange multipliers, no modifications in the reciprocal basis functions are needed to account for the boundary conditions. Hence, the present method does not require internal modifications of existing IGA software structures. Numerical examples show that globally continuous dual basis functions yield better accuracy than locally discontinuous biorthogonal functions, but with much higher computational overhead. Locally discontinuous dual basis functions are found to be an economical alternative to lumped mass matrices when combined with mass parameterization. The resulting inverse mass matrices are tested in several vibration problems and applied to explicit transient analysis of structures.

A computational approach for thermo-elasto-plastic frictional contact based on a monolithic formulation using non-smooth nonlinear complementarity functions

A new monolithic solution scheme for thermo-elasto-plasticity and thermo-elasto-plastic frictional contact with finite deformations and finite strains is presented. A key feature is the reformulation of all involved inequality constraints, namely those of Hill’s orthotropic yield criterion as well as the normal and tangential contact constraints, in terms of non-smooth nonlinear complementarity functions. Using a consistent linearization, this system of equations can be solved with a non-smooth variant of Newton’s method. A quadrature point-wise decoupled plastic constraint enforcement and the use of so-called dual basis functions in the mortar contact formulation allow for a condensation of all additionally introduced variables, thus resulting in an efficient formulation that contains discrete displacement and temperature degrees of freedom only, while, at the same time, an exact constraint enforcement is assured. Numerical examples from thermo-plasticity, thermo-elastic frictional contact and thermo-elasto-plastic frictional contact demonstrate the wide range of applications covered by the presented method.

Patch coupling with the isogeometric dual mortar approach

AbstractThe use of a common set of basis functions for design and analysis is the main paradigm of isogeometric analysis. The characteristics of the commonly used non‐uniform rational B‐splines (NURBS) surfaces require methods to handle non‐conforming meshes to attain an efficient computational framework. The isogeometric mortar method uses constrained approximation spaces to enforce a coupling of deformations at the interface between patches in a weak manner. This method neither requires additional degrees of freedom nor the choice of empirical parameters. The main drawback of the standard isogeometric mortar approach is the non‐local support of the mortar basis functions along the interface. This yields a large number of nodes per element for elements adjacent to the interface. Thus, the computational costs increase significantly for mesh refinement. This issue is remedied by the use of dual basis functions for the mortar method, which is referred to as dual mortar method. In this contribution several choices for the dual basis functions for B‐splines are proposed and compared. A special focus is set on the support of the dual basis functions and on the support of the resulting mortar basis functions. Numerical examples show the influence of the choice for the dual basis functions on the accuracy of the global stress distribution, on the fulfillment of the interface conditions and on numerical efficiency. The use of approximate dual basis functions is shown to be competitive to computations of conforming meshes in terms of accuracy and efficiency. (© 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Dual and approximate dual basis functions for B-splines and NURBS – Comparison and application for an efficient coupling of patches with the isogeometric mortar method

This contribution defines and compares different methods for the computation of dual basis functions for B-splines and Non-Uniform Rational B-splines (NURBS). They are intended to be used as test functions for the isogeometric mortar method, but other fields of application are possible, too. Three different concepts are presented and compared. The first concept is the explicit formula for the computation of dual basis functions for NURBS proposed in the work of Carl de Boor. These dual basis functions entail minimal support, i.e., the support of the dual basis functions is equal to the support of the corresponding B-spline basis functions. In the second concept dual basis functions are derived from the inversion of the Gram matrix. These dual basis functions have global support along the interface. The third concept is the use of approximate dual basis functions, which were initially proposed for the use in harmonic analysis. The support of these functions is local but larger than the support of the associated B-spline basis functions. We propose an extension of the approximate dual basis functions for NURBS basis functions. After providing the general formulas, we elaborate explicit expressions for several degrees of spline basis functions. All three approaches are applied in the frame of the mortar method for the coupling of non-conforming NURBS patches. A method which allows complex discretizations with multiple intersecting interfaces is presented. Numerical examples show that the explicitly defined dual basis functions with minimal support severely deteriorate the global stress convergence behavior of the mechanical analysis. This fact is in accordance with mathematical findings in literature, which state that the optimal reproduction degree of arbitrary functions is not possible without extending the support of the dual basis functions. The dual basis functions computed from the inverse of the Gram matrix yield accurate numerical results but the global support yields significantly higher computational costs in comparison to computations of conforming meshes. Only the approximate dual basis functions yield accurate and efficient computations, where neither accuracy nor efficiency is significantly deteriorated in comparison to computations of conforming meshes. All basic cases of T-intersections and star-intersections are studied. Furthermore, an example which combines all basic cases in a complex discretization is given. The applicability of the presented method for the nonlinear case and for shell formulations is shown with the help of one numerical example.

Isogeometric dual mortar methods for computational contact mechanics

In recent years, isogeometric analysis (IGA) has received great attention in many fields of computational mechanics research. Especially for computational contact mechanics, an exact and smooth surface representation is highly desirable. As a consequence, many well-known finite element methods and algorithms for contact mechanics have been transferred to IGA. In the present contribution, the so-called dual mortar method is investigated for both contact mechanics and classical domain decomposition using NURBS basis functions. In contrast to standard mortar methods, the use of dual basis functions for the Lagrange multiplier based on the mathematical concept of biorthogonality enables an easy elimination of the additional Lagrange multiplier degrees of freedom from the global system. This condensed system is smaller in size, and no longer of saddle point type but positive definite. A very simple and commonly used element-wise construction of the dual basis functions is directly transferred to the IGA case. The resulting Lagrange multiplier interpolation satisfies discrete inf–sup stability and biorthogonality, however, the reproduction order is limited to one. In the domain decomposition case, this results in a limitation of the spatial convergence order to O(h32) in the energy norm, whereas for unilateral contact, due to the lower regularity of the solution, optimal convergence rates are still met. Numerical examples are presented that illustrate these theoretical considerations on convergence rates and compare the newly developed isogeometric dual mortar contact formulation with its standard mortar counterpart as well as classical finite elements based on first and second order Lagrange polynomials.

Dual basis functions in subspaces of inner product spaces

Dual basis functions are well-studied in the literature for certain inner product spaces. In this paper, we introduce dual basis functions in subspaces of inner product spaces. The goal is to construct a basis for a subspace that is dual to a basis in a different subspace of the same dimension. This problem reduces to the standard dual basis problem when the two subspaces and bases are the same.The paper begins with a characterization and properties of dual basis in subspaces, including requirements for existence. Then, the construction is carried out for subspaces of the space of polynomials in the Bernstein basis. Two configurations are of particular interest: a symmetric case in which the dual basis is affine and converges to Lagrange polynomial interpolation, and an end-point case that converges to Hermite interpolation.

An abstract framework for a priori estimates for contact problems in 3D with quadratic finite elements

In this paper, a variationally consistent contact formulation is considered and we provide an abstract framework for the a priori error analysis in the special case of frictionless contact and small deformations. Special emphasis is put on quadratic mortar finite element methods. It is shown that under quite weak assumptions on the Lagrange multiplier space $${\mathcal{O} (h^{t-1}), 2 < t < \frac52}$$ , a priori results in the H 1-norm for the error in the displacement and in the H ?1/2-norm for the error in the surface traction can be established provided that the solution is regular enough. We discuss several choices of Lagrange multipliers ranging from the standard lowest order conforming finite elements to locally defined biorthogonal basis functions. The crucial property for the analysis is that the basis functions have a local positive mean value. Numerical results are exemplarily presented for one particular choice of biorthogonal (i.e. dual) basis functions and also comprise the case of finite deformation contact.