Biharmonic Equation Research Articles

Local behavior of solutions to a biharmonic equation with isolated singularity

This paper is concerned with positive classical solutions u of the biharmonic equation Δ2u+up=0 in the punctured ball B2﹨{0} of Rn, where n≥5 and p>(n+1)/(n−3). We classify the isolated singularities of positive solutions and describe the asymptotic behavior of positive singular solutions without the sign assumption for −Δu.

Lightning Stokes Solver

Gopal and Trefethen recently introduced “lightning solvers” for the 2D Laplace and Helmholtz equations, based on rational functions with poles exponentially clustered near singular corners. Making use of the Goursat representation in terms of analytic functions, we apply these methods to the biharmonic equation, specifically to 2D Stokes flow. Solutions to model problems are computed to 10-digit accuracy in less than a second of laptop time. As an illustration of the high accuracy, we resolve two or more counterrotating Moffatt eddies near a singular corner.

A mesh refinement scheme for fourth order bi-harmonic equation of mixed boundary-value elastic problems

Fourth order bi-harmonic equation is extensively used for stress-strain analysis of mixed boundary-value elastic problems. However, currently existing uniform mesh scheme based on finite difference method (FDM) needs vast amount of computational resources and efforts for an acceptable solution. Therefore, in this study, a mesh refinement (MR) scheme based on FDM is developed to solve fourth order bi-harmonic equation effectively. The developed MR scheme allows high resolution computation in sub-domains of interest and relatively low resolution in other regions which overcomes the memory exhausting problems associating with the traditional uniform mesh based FDM. In this paper, sub-domain that needs high resolution (mesh refinement) are identified based on gradient of stress and displacement vectors. A very high gradient in any region signifies the need of fine mesh because coarse grained meshes are not adequate to capture the sharply changing stresses or displacements. Once the sub-domains of interest are identified, the mesh refinement is done by splitting course meshes into smaller meshes. Several new stencils are created to satisfy the fourth order by harmonic equation and associated boundary conditions over the various sizes of meshes. The developed MR scheme has been applied to solve several classical mixed boundary-value elastic problems to show its applicability. In addition, the validity, effectiveness, and superiority of the MR scheme have been established by comparing of obtained solutions with uniform mesh results, finite element method (FEM) results, and the well-known analytical results. Our results show that the developed MR scheme can provide a more reliable and accurate result than the conventional uniform mesh scheme with a reduced number of equations, thus, saves a huge amount of computational memory.

Four Boundary Value Problems for a Nonlocal Biharmonic Equation in the Unit Ball

Solvability issues of four boundary value problems for a nonlocal biharmonic equation in the unit ball are investigated. Dirichlet, Neumann, Navier and Riquier–Neumann boundary value problems are studied. For the problems under consideration, existence and uniqueness theorems are proved. Necessary and sufficient conditions for the solvability of all problems are obtained and an integral representations of solutions are given in terms of the corresponding Green’s functions.

H-, p-, and hp-Versions of the Least-Squares Collocation Method for Solving Boundary Value Problems for Biharmonic Equation in Irregular Domains and Their Applications

h-, p-, and hp-Versions of the Least-Squares Collocation Method for Solving Boundary Value Problems for Biharmonic Equation in Irregular Domains and Their Applications

Poroelastic solution of a wellbore in a swelling rock with non-hydrostatic stress field

The stress distribution around a circular borehole has been studied extensively. The existing analytical poroelastic solutions, however, often neglect the complex interactions between the solid and fluid that can significantly affect the stress solution. This is important in unconventional gas reservoirs such as coalbeds and shale formations. In order to address this limitation, this paper presents the development of a poroelastic solution that takes into account the effect of gas sorption-induced strain. The solution considers drainage of the reservoir fluid through a vertical wellbore in an isotropic, homogenous, poroelastic rock with non-hydrostatic in situ stress field. The sorption-induced shrinkage of coal is modelled using a Langmuir-type curve which relates the volumetric change to the gas pressure. The redistributed stress field around the wellbore after depletion is found by applying Biot's definition of effective stress and Airey's stress functions, which leads to a solution of the inhomogeneous Biharmonic equation. Two sets of boundary conditions were considered in order to simulate the unsupported cavity (open-hole) and supported cavity (lined-hole) cases. The implementation was verified against a numerical solution for both open-hole and lined-hole cases. A comparative study was then conducted to show the significance of the sorption-induced shrinkage in the stress distribution. Finally, a parametric study analysed the sensitivity of the solution to different poroelastic parameters. The results demonstrate that the developed solution is a useful tool that can be employed alongside complex flow models in order to conduct efficient, field-scale coupled hydro-mechanical simulations, especially when the stress-dependent permeability of the reservoir is of concern.

A Generic Analytical Elastic Solution for Excavation Responses of an Arbitrarily Shaped Deep Opening under Biaxial In Situ Stresses

This paper presents a generic analytical solution for stresses and displacements developed around an arbitrarily shaped underground opening excavated in an elastic soil/rock mass with biaxial in situ stresses. A series of separable functions satisfying the governing biharmonic equation is employed as the basic stress function for the problem considered. With the basic stress function and the stress rotation equation, the normal stress and the shear stress on the opening boundary surface are expressed in terms of separable function series with unknown coefficients to make use of the boundary condition. Based on the variational theory, the unknown coefficients are determined by minimizing the difference between the real stress boundary condition and the assumed stress function on the opening surface. The excavation responses around different shapes of openings, including elliptical-shaped, ovaloid-shaped, horseshoe-shaped, and D-shaped openings, are investigated and compared with those from the finite-element simulations. The results show that both the stress field and the displacement field agree well with those from the corresponding numerical model, which verify the proposed solving technique. The proposed solution provides an easy, generic, and feasible approach to assess the stresses and displacements developed around an arbitrarily shaped opening, which is simpler and more straightforward than the complex theory–based solutions.

Monogenic functions with values in algebras of the second rank over the complex field and a generalized biharmonic equation with a triple characteristic

The statement that any two-dimensional algebra 𝔹* of the second rank with unity over the field of complex numbers contains such a basis {e1; e2} that 𝔹*-valued “analytic” functions Φ(xe1 + ye2) (x, y are real variables) satisfy such a fourth-order homogeneous partial differential equation with complex coefficients that its characteristic equation has a triple root is proved. A set of all triples (𝔹*; {e1; e2}; Φ) is described in the explicit form. A particular solution of this fourth-order partial differential equation is found by use of these “analytic” functions.

A weak divergence CDG method for the biharmonic equation on triangular and tetrahedral meshes

A conforming discontinuous Galerkin (CDG) C0-Pk finite element method is introduced for solving the biharmonic equation on triangular and tetrahedral meshes. A C0-Pk finite element function is a continuous and piecewise polynomial of degree k on a triangular or tetrahedral mesh. The CDG method is based on taking weak divergence on the gradient of C0-Pk finite elements. Optimal order error estimates in both a discrete H2 norm and the L2 norm are established. Numerical results are presented to verify the theory.

On the Biharmonic Heat Equation on Complete Riemannian Manifolds

We study entire solutions of the biharmonic heat equation on complete Riemannian manifolds without boundary. We provide exponential decay estimates for the biharmonic heat kernel under assumptions on the lower bound of Ricci curvature and noncollapsing of unit balls. And we prove a uniqueness criteria for the Cauchy problem. As corollaries, we prove the conservation law for the biharmonic heat kernel and a uniform \(L^\infty \) estimate for entire solutions starting with bounded initial data.

Anholonomik Koordinatlar Boyunca Yönlü Enerji Fonksiyonelleri

In this paper, a special case of directional energy functional is investigated by computing the directional energy and pseudoangle of unit vector fields in the ordinary three-dimensional space. This approach is also extended simultaneously to define the critical points of the directional energy functionals of the velocity fields. Then, the restriction of the harmonic maps and the extrema of the directional energy functionals is considered, Finally, we compute directional harmonic and biharmonic equations of the curvature vector fields to generalize total bending or energy of vector fields.

Multiple solutions for biharmonic critical Choquard equation involving sign-changing weight functions

Multiple solutions for biharmonic critical Choquard equation involving sign-changing weight functions

On Some Asymptotic Properties of Solutions to Biharmonic Equations

On Some Asymptotic Properties of Solutions to Biharmonic Equations

On the Cauchy Problem for the Biharmonic Equation

The work is devoted to the study of continuation and stability estimation of the solution of the Cauchy problem for the biharmonic equation in the domain G from its known values on the smooth part of the boundary @G. The problem under consideration belongs to the problems of mathematical physics in which there is no continuous dependence of solutions on the initial data. In this work, using the Carleman function, not only the biharmonic function itself, but also its derivatives are restored from the Cauchy data on a part of the boundary of the region. The stability estimates for the solution of the Cauchy problem in the classical sense are obtained

A weak Galerkin/finite difference method for time-fractional biharmonic problems in two dimensions

In this work, based on a finite difference scheme, we propose the weak Galerkin (WG) method for solving time-fractional biharmonic equations. Theoretically and numerically, the optimal error estimates for semi-discrete and fully discrete schemes have been investigated. Based on mathematical induction, stability is discussed for the fully discrete scheme that depends on the initial value and the source term. Numerical experiments are provided to confirm the theoretical claims made by the proposed schemes.

The Existence and Non-Existence of Sign-Changing Solutions to Bi-Harmonic Equations with a p-Laplacian

The Existence and Non-Existence of Sign-Changing Solutions to Bi-Harmonic Equations with a p-Laplacian

A Physics-Informed Neural Network Approach to Solution and Identification of Biharmonic Equations of Elasticity

We explore an application of the Physics-Informed Neural Networks (PINNs) in conjunction with Airy stress functions and Fourier series to find optimal solutions to a few reference biharmonic problems of elasticity and elastic plate theory. Biharmonic relations are fourth-order partial differential equations (PDEs) that are challenging to solve using classical numerical methods and have not been addressed using PINNs. Our work highlights a novel application of classical analytical methods to guide the construction of efficient neural networks with a minimal number of parameters that are very accurate and fast to evaluate. In particular, we find that enriching the feature space using Airy stress functions can significantly improve the accuracy of PINN solutions for biharmonic PDEs.

Extraction of stress intensity factors of biharmonic equations with corner singularities corresponding to mixed boundary conditions of clamped, simply supported, and free (II)

In [13], we derived stress intensity factors (SIF) extraction formulas of a biharmonic equation Δ2u=f in a cracked domain whose crack faces have clamped boundary conditions. In this paper, we extend our investigation to the derivation of the SIF extraction formulas of the biharmonic equation in non-convex polygonal domains containing cracks or reentrant corners when various BC such as clamped (CC), simply supported (SS), free (FF), or mixed conditions (CS, FC, SF) are imposed on the boundary. We prove that the SIF is expressed as the integral of fΨsk⁎−uΔ2Ψsk⁎ for a cut-off function Ψ and a dual singular function sk⁎ on a small neighborhood of the singularity. The dual singular function is determined in this paper. For a numerical approximation of u, we proposed an iteration method as well as a direct finite element method. For a finite element solution of Δ2u=f, we have to use C1-continuous basis functions. For continuous differentiable basis functions, we use either C1-continuous B-spline basis functions or the conventional Hermite basis function. Because of several advantages of B-spline functions in imposing complex boundary conditions, we choose B-spline basis functions for the finite element approximation of u. Moreover, in the direct numerical method, we propose to use implicitly enriched basis functions that resemble the singularities.

The index of some mixed order Dirac type operators and generalised Dirichlet–Neumann tensor fields

AbstractWe revisit a construction principle of Fredholm operators using Hilbert complexes of densely defined, closed linear operators and apply this to particular choices of differential operators. The resulting index is then computed using an explicit description of the cohomology groups of generalised (‘harmonic’) Dirichlet and Neumann tensor fields. The main results of this contribution are the computation of the indices of Dirac type operators associated to the elasticity complex and the newly found biharmonic complex, relevant for the biharmonic equation, elasticity, and for the theory of general relativity. The differential operators are of mixed order and cannot be seen as leading order type with relatively compact perturbation. As a by-product we present a comprehensive description of the underlying generalised Dirichlet–Neumann vector and tensor fields defining the respective cohomology groups, including an explicit construction of bases in terms of topological invariants, which are of both analytical and numerical interest. Though being defined by certain projection mechanisms, we shall present a way of computing these basis functions by solving certain PDEs given in variational form. For all of this we rephrase core arguments in the work of Rainer Picard [42] applied to the de Rham complex and use them as a blueprint for the more involved cases presented here. In passing, we also provide new vector-analytical estimates of generalised Poincaré–Friedrichs type useful for elasticity or the theory of general relativity.

Existence of solution to a nonlocal biharmonic problem with dependence on gradient and Laplacian

Abstract In this article, we prove the existence of a solution to a nonlocal biharmonic equation with nonlinearity depending on the gradient and the Laplacian. We employ an iterative technique based on the mountain pass theorem to prove our result.