Interior Nodal Research Articles

An adaptive C0 Interior Penalty Discontinuous Galerkin method and an equilibrated a posteriori error estimator for the von Kármán equations

We consider an adaptive C0 Interior Penalty Discontinuous Galerkin (C0IPDG) approximation of the fourth order von Kármán equations with homogeneous Dirichlet boundary conditions and an equilibrated a posteriori error estimator. The C0IPDG method can be derived from a six-field formulation of the finite element discretized von Kármán equations. The equilibrated a posteriori error estimator provides an upper bound for the discretization error in the broken W02,2 norm in terms of the associated primal and dual energy functionals. It requires the construction of equilibrated fluxes and equilibrated moment tensors which can be computed on local patches around interior nodal points of the triangulations. The relationship with a residual-type a posteriori error estimator is studied as well. Numerical results illustrate the performance of the suggested approach.

Geometry and interior nodal sets of Steklov eigenfunctions

We investigate the geometric properties of Steklov eigenfunctions in smooth manifolds. We derive the refined doubling estimates and Bernstein’s inequalities. For the real analytic manifolds, we are able to obtain the sharp upper bound for the measure of interior nodal sets.

Polynomial upper bound on interior Steklov nodal sets

We study solutions of uniformly elliptic PDE with Lipschitz leading coefficients and bounded lower order coefficients. We extend previous results of A. Logunov ([9]) concerning nodal sets of harmonic functions and, in particular, prove polynomial upper bounds on interior nodal sets of Steklov eigenfunctions in terms of the corresponding eigenvalue \lambda .

THE TREFFTZ TEST FUNCTIONS METHOD FOR SOLVING THE GENERALIZED INVERSE BOUNDARY VALUE PROBLEMS OF LAPLACE EQUATION

The issue of data completion is important for the elliptic type partial differential equation. In the inverse Cauchy problem, we need to complete the boundary data by over-specifying Dirichlet and Neumann data on a portion of the boundary. In this paper, we numerically solve the generalized inverse boundary value problems of Laplace equation in a rectangle with one boundary function and two boundary functions missing, which are more difficult than the inverse Cauchy problem. By using the technique of a boundary integral equation method together with a specially designed Trefftz test function, we can complete the boundary data by requiring minimal extra data. Then solving the Laplace equation with the given data and recovered data by the multiple-scale Trefftz method, we can find the numerical solution in the interior nodal points.

Lower bounds for nodal sets of biharmonic Steklov problems

We use layer potential to establish that the boundary biharmonic Steklov operators are elliptic pseudo-differential operators. Thus we are able to establish lower bounds on both the measure of boundary nodal sets and interior nodal sets for biharmonic Steklov eigenfunctions.

Lower bounds for interior nodal sets of Steklov eigenfunctions

We study the interior nodal sets, Z λ Z_\lambda of Steklov eigenfunctions in an n n -dimensional relatively compact manifold M M with boundary and show that one has the lower bounds | Z λ | ≥ c λ 2 − n 2 |Z_\lambda |\ge c\lambda ^{\frac {2-n}2} for the size of its ( n − 1 ) (n-1) -dimensional Hausdorff measure. The proof is based on a Dong-type identity and estimates for the gradient of Steklov eigenfunctions, similar to those in previous works of the first author and Zelditch.

Interior nodal sets of Steklov eigenfunctions on surfaces

We investigate the interior nodal sets $\mathcal{N}_\lambda$ of Steklov eigenfunctions on connected and compact surfaces with boundary. The optimal vanishing order of Steklov eigenfunctions is shown be $C\lambda$. The singular sets $\mathcal{S}_\lambda$ are finite points on the nodal sets. We are able to prove that the Hausdorff measure $H^0(\mathcal{S}_\lambda)\leq C\lambda^2$. Furthermore, we obtain an upper bound for the measure of interior nodal sets $H^1(\mathcal{N}_\lambda)\leq C\lambda^{\frac{3}{2}}$. Here those positive constants $C$ depend only on the surfaces.

Inverse problems for the boundary value problem with the interior nodal subsets

In this paper, inverse nodal problems for Sturm–Liouville equations with boundary conditions polynomially dependent on the spectral parameter were studied. The authors showed that some uniqueness theorems on the potential function hold by the Weyl function, respectively.

Sign-changing solutions of Lane Emden problems with interior nodal line and semilinear heat equations

We consider the semilinear Lane Emden problem{−Δu=|u|p−1uin Ω,u=0on ∂Ω where Ω is a smooth bounded simply connected domain in R2, invariant by the action of a finite symmetry group G.We show that if the orbit of each point in Ω, under the action of the group G, has cardinality greater than or equal to 4 then, for p sufficiently large, there exists a sign-changing solution of (Ep) with two nodal regions whose nodal line does not touch ∂Ω.This result is proved as a consequence of an analogous result for the associated parabolic problem.

A robust solution procedure for hyperelastic solids with large boundary deformation

Compressible Mooney–Rivlin theory has been used to model hyperelastic solids, such as rubber and porous polymers, and more recently for the modeling of soft tissues for biomedical tissues, undergoing large elastic deformations. We propose a solution procedure for Lagrangian finite element discretization of a static nonlinear compressible Mooney–Rivlin hyperelastic solid. We consider the case in which the boundary condition is a large prescribed deformation, so that mesh tangling becomes an obstacle for straightforward algorithms. Our solution procedure involves a largely geometric procedure to untangle the mesh: solution of a sequence of linear systems to obtain initial guesses for interior nodal positions for which no element is inverted. After the mesh is untangled, we take Newton iterations to converge to a mechanical equilibrium. The Newton iterations are safeguarded by a line search similar to one used in optimization. Our computational results indicate that the algorithm is up to 70 times faster than a straightforward Newton continuation procedure and is also more robust (i.e., able to tolerate much larger deformations). For a few extremely large deformations, the deformed mesh could only be computed through the use of an expensive Newton continuation method while using a tight convergence tolerance and taking very small steps.

On symmetry of nonnegative solutions of elliptic equations

We consider the Dirichlet problem for a class of fully nonlinear elliptic equations on a bounded domain Ω. We assume that Ω is symmetric about a hyperplane H and convex in the direction perpendicular to H. By a well-known result of Gidas, Ni and Nirenberg and its generalizations, all positive solutions are reflectionally symmetric about H and decreasing away from the hyperplane in the direction orthogonal to H. For nonnegative solutions, this result is not always true. We show that, nonetheless, the symmetry part of the result remains valid for nonnegative solutions: any nonnegative solution u is symmetric about H. Moreover, we prove that if u≢0, then the nodal set of u divides the domain Ω into a finite number of reflectionally symmetric subdomains in which u has the usual Gidas–Ni–Nirenberg symmetry and monotonicity properties. We also show several examples of nonnegative solutions with a nonempty interior nodal set.

Polynomial spline approach for solving second-order boundary-value problems with Neumann conditions

In this paper, a new difference scheme based on quartic splines is derived for solving linear and nonlinear second-order ordinary differential equations subject to Neumann-type boundary conditions. The scheme can achieve sixth order accuracy at the interior nodal points and fourth order accuracy at and near the boundary, which is superior to the well-known Numerov’s scheme with the accuracy being fourth order. Convergence analysis of the present method for linear cases is discussed. Finally, numerical results for both linear and nonlinear cases are given to illustrate the efficiency of our method.

Local integral equations implemented by MLS-approximation and analytical integrations

This paper is devoted to the development of advanced mesh free implementations of the governing equations and the boundary conditions for boundary value problems in elasticity. Both the strong and weak formulations are discretized by using the Moving Least Squares approximations. The weak formulation is represented by local integral equations considered on sub-domains around interior nodal points. The awkward evaluation of the shape functions and their derivatives is reduced by focusing to nodal points because of the development of analytical integrations. That results in significant saving of the computational time needed for creation of the system matrix. Furthermore, a modified differentiation scheme is developed for approximation of higher order derivatives of displacements appearing in the discretized formulations. The accuracy, convergence and computational efficiency are studied in simple numerical example.

Investigation of elastic field of a short orthotropic composite column by using finite-difference technique

The elastic field of a short column of orthotropic composite material under a linearly varying distributed load is analysed using displacement potential formulation. The finite-difference technique is used to obtain the numerical solutions of different parameters of the elastic field. The differential equations associated with the boundary conditions and the governing equation are approximated by their corresponding finite-difference forms, which are then applied, respectively, to the boundaries and the interior nodal points of the discretized column. To demonstrate the elastic behaviour of the fibre reinforced composite column, the results of displacements and stresses at different sections are presented in a graphical form.

Transient heat conduction analysis by triple-reciprocity boundary element method

The paper deals with 2D initial-boundary value problems in the linear theory of transient heat conduction. A pure boundary element formulation is developed systematically. The time-dependent fundamental solution of the diffusion operator is employed together with higher-order polyharmonic fundamental solutions. The pseudo-initial temperature and/or heat sources density are approximated by using the triple-reciprocity formulation. All the time integrations are performed analytically in the time-marching scheme with integration within one time step and constant interpolation. The spatial discretization is reduced to boundary elements and free scattering of interior nodal points without any connectivity.

Algorithms for automatic generating interior nodal points and Delaunay triangulation using advancing front technique

AbstractA new algorithm is proposed for generating interior nodal points within an arbitrary two‐dimensional domain. The algorithm is based on a background triangle mesh and the contours on the background mesh. The newly generated points are exactly within the domain with smooth point densities and a good quality of the final mesh. An improved Delaunay triangulation algorithm using advancing front technique is also proposed. The present algorithm is more general and robust. A FORTRAN 90 program based on the present algorithms is developed and has been successfully applied in many projects. Examples are given to show the effectiveness and robustness of the algorithms. Copyright © 2005 John Wiley & Sons, Ltd.

An advanced numerical method for computing elastodynamic fracture parameters in functionally graded materials

A new meshless method for computing the dynamic stress intensity factors (SIFs) in continuously non-homogeneous solids under a transient dynamic load is presented. The method is based on the local boundary integral equation (LBIE) formulation and the moving least squares (MLS) approximation. The analyzed domain is divided into small subdomains, in which a weak solution is assumed to exist. Nodal points are randomly spread in the analyzed domain and each one is surrounded by a circle centered at the collocation point. The boundary-domain integral formulation with elastostatic fundamental solutions for homogeneous solids in Laplace-transformed domain is used to obtain the weak solution for subdomains. On the boundary of the subdomains, both the displacement and the traction vectors are unknown generally. If modified elastostatic fundamental solutions vanishing on the boundary of the subdomain are employed, the traction vector is eliminated from the local boundary integral equations for all interior nodal points. The spatial variation of the displacements is approximated by the MLS scheme.

A boundary integral equation method using auxiliary interior surface approach for acoustic radiation and scattering in two dimensions.

This paper presents an effective solution method for predicting acoustic radiation and scattering fields in two dimensions. The difficulty of the fictitious characteristic frequency is overcome by incorporating an auxiliary interior surface that satisfies certain boundary condition into the body surface. This process gives rise to a set of uniquely solvable boundary integral equations. Distributing monopoles with unknown strengths over the body and interior surfaces yields the simple source formulation. The modified boundary integral equations are further transformed to ordinary ones that contain nonsingular kernels only. This implementation allows direct application of standard quadrature formulas over the entire integration domain; that is, the collocation points are exactly the positions at which the integration points are located. Selecting the interior surface is an easy task. Moreover, only a few corresponding interior nodal points are sufficient for the computation. Numerical calculations consist of the acoustic radiation and scattering by acoustically hard elliptic and rectangular cylinders. Comparisons with analytical solutions are made. Numerical results demonstrate the efficiency and accuracy of the current solution method.

SUBSTRUCTURAL IDENTIFICATION OF BRIDGE

The purpose of this paper is to use the seismic response data of one 5-span continuous box-girder bridge to identify the dynamic characteristics of the system from different earthquake loading. The spectral finite element method was used by considering the flexural wave in beam to form the dynamic stiffness matrix of the substructure. Through matrix condensation the interior nodal displacement can be expressed in terms of the boundary nodal displacements. The identification was performed by using the recorded nodal displacement of the bridge to identify the EI-value and damping factor η of the substructure. Three seismic event data sets were used from the bridge strong motion instrumentation array. It is found that the identified EI-values are very stable even for different intensity of earthquake loading while the damping value may change from seismic event to event.

Stress intensity factors for two-dimensional crack problems using constrained finite elements

Two-dimensional crack problems, in common with other elliptic problems containing a boundary singularity, may be solved efficiently with the aid of a constrained finite element. The singularity is surrounded by a superelement containing a refined mesh whose interior nodal values are constrained to agree with the first few terms of the known expansion for the solution. The superelement conforms with linear or bilinear elements, and may thus be included in standard finite element programs. The calculation yields the expansion coefficients directly, and the method has been applied to determine stress intensity factors for a variety of two-dimensional configurations, including mixed-mode. The results are in excellent agreement with those obtained by other methods.