Fourth-order Problem Research Articles

Kansa radial basis function method with fictitious centres for solving nonlinear boundary value problems

A Kansa–radial basis function (RBF) collocation method is applied to two–dimensional second and fourth order nonlinear boundary value problems. The solution is approximated by a linear combination of RBFs, each of which is associated with a centre and a different shape parameter. As well as the RBF coefficients in the approximation, these shape parameter values are taken to be among the unknowns. In addition, the centres are distributed within a larger domain containing the physical domain of the problem. The size of this larger domain is controlled by a dilation parameter which is also included in the unknowns. In fourth order problems where two boundary conditions are imposed, two sets of (different) boundary centres are selected. The Kansa–RBF discretization yields a system of nonlinear equations which is solved by standard software. The proposed technique is applied to four problems and the numerical results are analyzed and discussed.

Optimal-rate finite-element solution of Dirichlet problems in curved domains with straight-edged tetrahedra

Abstract In a series of papers published since 2017 the author introduced a simple alternative of the $n$-simplex type, to enhance the accuracy of approximations of second-order boundary value problems subject to Dirichlet boundary conditions, posed on smooth curved domains. This technique is based upon trial functions consisting of piecewise polynomials defined on straight-edged triangular or tetrahedral meshes, interpolating the Dirichlet boundary conditions at points of the true boundary. In contrast, the test functions are defined by the standard degrees of freedom associated with the underlying method for polytopic domains. While the mathematical analysis of the method for Lagrange and Hermite methods for two-dimensional second- and fourth-order problems was carried out in earlier paper by the author this paper is devoted to the study of the three-dimensional case. Well-posedness, uniform stability and optimal a priori error estimates in the energy norm are proved for a tetrahedron-based Lagrange family of finite elements. Novel error estimates in the $L^2$-norm, for the class of problems considered in this work, are also proved. A series of numerical examples illustrates the potential of the new technique. In particular, its superior accuracy at equivalent cost, as compared to the isoparametric technique, is highlighted.

Beams with an intermediate pier: Spectral properties, asymmetry and stability

We deal with beams with an intermediate pier, motivated by the investigation of the stability of suspension bridges with two spans. First, we provide a complete spectral theorem for the associated linear stationary fourth-order problem with hinged boundary conditions, determining the eigenvalues and discussing their optimality (in a suitable sense) in terms of the position of the pier. Then, we consider a related nonlinear model with a restoring force of superquadratic displacement type, discussing its stability both from a linear and from a suitable nonlinear point of view. We determine the position of the pier maximizing the stability of the structure and we compare the energy thresholds of instability under hinged, clamped or mixed (left-clamped and right-hinged) boundary conditions. In any case, we highlight that an asymmetric structure is in general more stable.

Stationary solutions to a nonlocal fourth-order elliptic obstacle problem

Existence of stationary solutions to a nonlocal fourth-order elliptic obstacle problem arising from the modelling of microelectromechanical systems with heterogeneous dielectric properties is shown. The underlying variational structure of the model is exploited to construct these solutions as minimizers of a suitably regularized energy, which allows us to weaken considerably the assumptions on the model used in a previous article.

Block Hybrid Method for the Numerical solution of Fourth order Boundary Value Problems

A linear Multistep Block Hybrid Method with four off-grid points is presented for the direct approximation of the solution of fourth order Boundary Value Problems. Multiple Finite Difference formulas are derived and grouped into a unique block to form a numerical integrator to solve directly the fourth order problem, without the need to reduce it previously to a first-order system. The convergence of the proposed method is discussed. The superiority of this method over existing methods is established numerically considering different problems that have appeared in the literature.

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.

Spectral Method and Spectral Element Method for Three Dimensional Linear Elliptic System: Analysis and Application

In this paper, a least-squares spectral method and a non-conforming least-squares spectral element method for three dimensional linear elliptic system will be presented. Differentiability estimates and the main stability theorem for the proposed method are proven. Using the regularity estimate and the proposed stability estimates, we introduce a suitable preconditioner and show that the condition number of the preconditioned system is O((ln W)^2) (where W is degree of polynomials). Then we establish the error estimates of the proposed method. Specific numerical results based on linear elasticity problem, fourth order problem and sixth order problem are discussed to reflect the efficiency of the proposed method.

Positive Solutions of Fourth-Order Problem Subject to Nonlocal Boundary Conditions

In this paper, we study the fourth-order problem with the second derivative in nonlinearity under nonlocal boundary value conditions involving Stieltjes integrals. Some inequality conditions on nonlinearity and the spectral radius conditions of linear operators are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on special cone. The conditions allow that the nonlinearity has superlinear or sublinear growth. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel.

Eigenvibrations of a beam with two mechanical resonators attached to the ends

The fourth-order ordinary differential spectral problem describing vertical eigenvibrations of a beam with two mechanical resonators attached to the ends is studied. This problem has positive simple eigenvalues and corresponding eigenfunctions. We define limit differential spectral problem and establish the convergence of the eigenvalues and eigenfunctions of the original spectral problem to the eigenvalues and eigenfunctions of the limit spectral problem as parameters of the attached resonators tending to infinity. The initial fourth-order ordinary differential spectral problem is approximated by the finite difference method. Theoretical error estimates for approximate eigenvalues and eigenfunctions are derived. Obtained theoretical results are illustrated by computations for model problem with constant coefficients. Theoretical and experimental results of this paper can be developed for the problems on eigenvibrations of complex mechanical constructions with systems of resonators.

Code Generation for Generally Mapped Finite Elements

Many classical finite elements such as the Argyris and Bell elements have long been absent from high-level PDE software. Building on recent theoretical work, we describe how to implement very general finite-element transformations in FInAT and hence into the Firedrake finite-element system. Numerical results evaluate the new elements, comparing them to existing methods for classical problems. For a second-order model problem, we find that new elements give smooth solutions at a mild increase in cost over standard Lagrange elements. For fourth-order problems, however, the newly enabled methods significantly outperform interior penalty formulations. We also give some advanced use cases, solving the nonlinear Cahn-Hilliard equation and some biharmonic eigenvalue problems (including Chladni plates) using C 1 discretizations.

Diagonalized Gegenbauer rational spectral methods for second- and fourth-order problems on the whole line

Fully diagonalized Gegenbauer rational spectral methods for solving second- and fourth-order differential equations on the whole line are proposed and analyzed. Some Gegenbauer rational Sobolev orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions can be represented as infinite and truncated Gegenbauer rational series. Optimal error estimates of the fully diagonalized Gegenbauer rational spectral method for second-order problem are obtained. Finally, some numerical experiments, which are in agreement with the theoretical analysis, demonstrate the effectiveness and the spectral accuracy of our diagonalized methods.

Linear theory for beams with intermediate piers

The full linear theory for hinged beams with intermediate piers is developed. The analysis starts with the variational setting and the study of the linear stationary problem. Well-posedness results are provided and the possible loss of regularity, due to the presence of the piers, is analyzed. A complete spectral theorem is then proved, explicitly determining the eigenvalues according to the position of the piers and exhibiting the fundamental modes of oscillation. A related second-order eigenvalue problem is also studied, showing that it may display nonsmooth eigenfunctions and that the fourth-order problem cannot be seen as the square of a second-order problem.

Computation of interior elastic transmission eigenvalues using a conforming finite element and the secant method

The interior elastic transmission eigenvalue problem, arising from the inverse scattering theory of non-homogeneous elastic media, is nonlinear, non-self-adjoint and of fourth order. This paper proposes a numerical method to compute real elastic transmission eigenvalues. To avoid treating the non-self-adjoint operator, an auxiliary nonlinear function is constructed. The values of the function are generalized eigenvalues of a series of self-adjoint fourth order problems. The roots of the function are the transmission eigenvalues. The self-adjoint fourth order problems are computed using the H2-conforming Argyris element. The secant method is employed to search the roots of the nonlinear function. The convergence of the proposed method is proved.

Multiplicity results for fourth order problems related to the theory of deformations beams

The main purpose of this paper is to establish the existence and multiplicity of positive solutions for a fourth-order boundary value problem with integral condition. By using a new technique of construct a positive cone, we apply the Krasnoselskii compression/expansion and Leggett-Williams fixed point theorems in cones to show our multiplicity results. Finally, a particular case is studied, and the existence of multiple solutions is proved for two different particular functions.

Numerical computations of split Bregman method for fourth order total variation flow

The split Bregman framework for Osher-Solé-Vese (OSV) model and fourth order total variation flow are studied. We discretize the problem by piecewise constant function and compute ∇(−Δav)−1 approximately and exactly. Furthermore, we provide a new shrinkage operator for Spohn's fourth order model. Numerical experiments are demonstrated for fourth order problems under periodic boundary condition.

Oscillation Properties of a Multipoint Fourth-Order Boundary-Value Problem with Spectral Parameter in the Boundary Condition

A multipoint fourth-order boundary-value problem with spectral parameter in the boundary condition is considered. It is proved that its spectrum is simple and the system of derivative eigenfunctions has oscillation properties.

A [formula omitted] Virtual Element Method on polyhedral meshes

The purpose of the present paper is to develop C1 Virtual Elements in three dimensions for linear elliptic fourth order problems, motivated by the difficulties that standard conforming Finite Elements encounter in this framework. We focus the presentation on the lowest order case, the generalization to higher orders being briefly provided in the Appendix. The degrees of freedom of the proposed scheme are only 4 per mesh vertex, representing function values and gradient values. Interpolation error estimates for the proposed space are provided, together with a set of numerical tests to validate the method at the practical level.

Properties of the Fourth-Order Boundary Value Problem with Transmission Conditions

We investigate the fourth-order eigenvalue problem with transmission conditions. We apply the Fourier method for solving of this problem arising in finding of transverse vibration of a string. We derive the eigenvalues and eigenfunctions of Sturm–Liouville problems and their properties. We obtain a solution of the considered equation subject to initial, boundary and transmission conditions at the point of discontinuity.

Eigenvalues of a class of fourth-order boundary value problems with transmission conditions using matrix theory

ABSTRACT A class of fourth-order boundary value problems with general self-adjoint boundary and transmission conditions is investigated. It is shown that such boundary value problems with a finite number of eigenvalues are equivalent to certain types of matrix eigenvalue problems. Moreover, the results are given under the general self-adjoint transmission conditions and the self-adjoint canonical forms of the fourth-order problems which are classified as separated, mixed and coupled, respectively.

High-precision computation of the weak Galerkin methods for the fourth-order problem

The weak Galerkin form of the finite element method, requiring only C0 basis function, is applied to the biharmonic equation. The computational procedure is thoroughly considered. Local orthogonal bases on triangulations are constructed using various sets of interpolation points with the Gram-Schmidt or Levenberg-Marquardt methods. Comparison and high-precision computations are carried out, and convergence rates are provided up to degree 11 for L2, 10 for H1, and 9 for H2, suggesting that the algorithm is useful for a variety of computations.