Boundary Value Problems Of Elasticity Research Articles

Linel2D-Net: A deep learning approach to solving 2D linear elastic boundary value problems on image domains

Efficient solution of physical boundary value problems (BVPs) remains a challenging task demanded in many applications. Conventional numerical methods require time-consuming domain discretization and solving techniques that have limited throughput capabilities. Here, we present an efficient data-driven DNN approach to non-iterative solving arbitrary 2D linear elastic BVPs. Our results show that a U-Net-based surrogate model trained on a representative set of reference FDM solutions can accurately emulate linear elastic material behavior with manifold applications in deformable modeling and simulation.

Analysis of plane elasticity problems using the dual mesh control domain method

The dual mesh control domain method (DMCDM) proposed by Reddy combines the advantages of the finite element method (interpolation of variables) and the finite volume method (satisfaction of the global form of the governing equations). Currently, the method has been used to solve two-dimensional heat transfer, fluid and solid mechanics problems on structured meshes of linear right triangular and bilinear rectangular elements. However, such meshes are not useful from the standpoint of most practical problems which are posed on irregular domains. The present study extends the DMCDM for the analysis of plane elasticity problems involving non-rectangular domains meshed with arbitrarily shaped elements. The approach makes use of the isoparametric formulation, in which the primal mesh used for the approximation of the geometry is also used for approximating the dependent unknowns. Several boundary value problems of elasticity are solved using the DMCDM, and the results are compared with those obtained using the finite element method (FEM). The results show that the DMCDM exhibits excellent convergence and accuracy while requiring less computational cost when compared with the FEM.

Node's residual descent method for linear elastic boundary value problems

This study proposes a node's residual descent method (NRDM) for the linear elasticity boundary value problem. In this method, the boundary information and deformation states are attached to the nodes, and the partial differential equations (PDEs) are then transformed into a generalized difference form with the first-order generalized finite difference (GFD) scheme. Furthermore, this study thoroughly describes the NRDM, confirms the accuracy and convergence by numerical examples, performs an error analysis, and discusses the influence of computational parameters. Results demonstrate that NRDM has high computational accuracy and superliner convergence rate. The double derivation is implemented to solve the second-order PDE boundary value problem with the first-order GFD scheme, thus simplifying the requirement for star connectivity and reducing the computational complexity. NRDM is a matrix-free iterative method that eliminates the task of assembling large global sparse matrices, thus avoiding the numerous matrix–matrix product operations. Rather than solving large systems of linear equations, many extremely small matrix multiplications are present during iterations, which are extremely friendly to parallel computing and can be adapted to the computation of ultra-multi-degree-of-freedom problems. The computational accuracy and convergence rate are considerably improved in NRDM compared to the classical GFD method.

Kernel free boundary integral method for 3D incompressible flow and linear elasticity equations on irregular domains

A second-order accurate kernel-free boundary integral method is presented for Stokes flow and linear elasticity boundary value problems on three-dimensional irregular domains. It solves equations in the framework of boundary integral equations, whose corresponding discrete forms are well-conditioned and solved by the GMRES method. A notable feature of this approach is that the boundary or volume integrals encountered in BIEs are indirectly evaluated by a Cartesian grid-based method, which includes discretizing corresponding simple interface problems with a MAC scheme, correcting discrete linear systems to reduce large local truncation errors near the interface, solving the modified system by a CG method together with an FFT-based Poisson solver. No extra work or special quadratures are required to deal with singular or hyper-singular boundary integrals and the dependence on the analytical expressions of Green’s functions for the integral kernels is completely eliminated. Numerical results are given to demonstrate the efficiency and accuracy of the Cartesian grid-based method.

Two-Term Spectral Asymptotics in Linear Elasticity

We establish the two-term spectral asymptotics for boundary value problems of linear elasticity on a smooth compact Riemannian manifold of arbitrary dimension. We also present some illustrative examples and give a historical overview of the subject. In particular, we correct erroneous results published by Liu (J Geom Anal 31:10164–10193, 2021).

Discontinuous Mappings and the Limit Load in Boundary Value Problems of Nonlinear Elasticity

Discontinuous Mappings and the Limit Load in Boundary Value Problems of Nonlinear Elasticity

Lithospheric Stress Due to Mantle Convection and Mantle Plume over East Africa from GOCE and Seismic Data

The Afar and Ethiopian plateaus are in a dynamic uplift due to the mantle plume, therefore, considering the plume effect is necessary for any geophysical investigation including the estimation of lithospheric stress in this area. The Earth gravity models of the Gravity Field and Steady-State Ocean Circulation Explorer (GOCE) and lithospheric structure models can be applied to estimate the stress tensor inside the Ethiopian lithosphere. To do so, the boundary-value problem of elasticity is solved to derive a general solution for the displacement field in a thin elastic spherical shell representing the lithosphere. After that, general solutions for the elements of the strain tensor are derived from the displacement field, and finally the stress tensor from the strain tensor. The horizontal shear stresses due to mantle convection and the vertical stress due to the mantle plume are taken as the lower boundary value at the base of the lithosphere, and no stress at the upper boundary value of the lithospheric shell. The stress tensor and maximum stress directions are computed at the Moho boundary in three scenarios: considering horizontal shear stresses due to mantle convection, vertical stresses due to mantle plume, and their combination. The estimated maximum horizontal shear stresses’ locations are consistent with tectonics and seismic activities in the study area. In addition, the maximum shear stress directions are highly correlated with the World Stress Map 2016, especially when the effect of the mantle plume is solely considered, indicating the stress in the study area mainly comes from the plume.

Discontinuous mappings and the limit load in boundary value problems of nonlinear elasticity

The article discusses a boundary value problem of nonlinear elasticity for mapping (deformation) in two weak formulations: in the form of a variational equilibrium equation and in the form of minimization of a multidimensional integral functional. Some questions of a mathematical correctness of the nonlinear elasticity BVP are discussed. Using methods of calculus of variations, by the example of two simple problems, it is proved that for some nonlinear elastic models in the corresponding boundary value problems there can be mappings that have discontinuities of the slip type, as well as there is the limit load - such a finite value of external forces, above which the boundary value problem has no solution at all.

A Formal Solution of Quadruple Series Equations

It cannot be overstated how significant Series Equations are to the fields of pure and applied mathematics respectively. The majority of mathematical topics revolve around the use of series. Virtually, in every subject of mathematics, series play an important role. Series solutions play a major role in the solution of mixed boundary value problems. Dual, triple, and quadruple series equations are useful in finding the solution of four part boundary value problems of electrostatics, elasticity and other fields of Mathematical physics. Cooke devised a method for finding the solution of quadruple series equations involving Fourier-Bessel series and obtained the solution using operator theory. Several workers have devoted considerable attention to the solutions of various equations involving for instance, trigonometric series, The Fourier-Bessel series, The Fourier Legendre series, The Dini series, series of Jacobi and Laguerre polynomials and series equations involving Bateman K-functions. Indeed, many of these problems arise in the investigation of certain classes of mixed boundary value problems in potential theory. There has been less work on quadruple series equations involving various polynomials and functions. In light of the significance of quadruple series solutions, proposed work examines quadruple series equations that include the product of r generalised Bateman K functions. Solution is formal, and there has been no attempt made to rationalise many restricting processes that have been encountered.

Numerical method for solving of ill-conditioned boundary value problems in nonlinear elasticity

Numerical method for solving of ill-conditioned boundary value problems in nonlinear elasticity

H-Adaptive radial basis function finite difference method for linear elasticity problems

In this research work, the radial basis function finite difference method (RBF-FD) is further developed to solve one- and two-dimensional boundary value problems in linear elasticity. The related differentiation weights are generated by using the extended version of the RBF utilizing a polynomial basis. The type of the RBF is restricted to polyharmonic splines (PHS), i.e., a combination of the odd m-order PHS phi (r)=r^m with additional polynomials up to degree p will serve as the basis. Furthermore, a new residual-based adaptive point-cloud refinement algorithm will be presented and its numerical performance will be demonstrated. The computational efficiency of the PHS RBF-FD approach is tested by means of the relative errors measured in ell _2-norm on several representative benchmark problems with smooth and non-smooth solutions, using h-adaptive, uniform, and quasi-uniform point-cloud refinement.

ANALYTICAL SOLUTION OF THE HILBERT SINGULAR INTEGRAL EQUATION WITH RESPECT TO THE HILBERT FORMULA

The objective of this paper is to solve the Hilbert singular integral equation of first and second kind by using the Hilbert formula and some known results are obtained as special cases. Applications of these singular integral equations to Boundary value problems of Elasticity and allied subjects are well known.

Data-driven tissue mechanics with polyconvex neural ordinary differential equations

Data-driven methods are becoming an essential part of computational mechanics due to their advantages over traditional material modeling. Deep neural networks are able to learn complex material response without the constraints of closed-form models. However, data-driven approaches do not a priori satisfy physics-based mathematical requirements such as polyconvexity, a condition needed for the existence of minimizers for boundary value problems in elasticity. In this study, we use a recent class of neural networks, neural ordinary differential equations (N-ODEs), to develop data-driven material models that automatically satisfy polyconvexity of the strain energy. We take advantage of the properties of ordinary differential equations to create monotonic functions that approximate the derivatives of the strain energy with respect to deformation invariants. The monotonicity of the derivatives guarantees the convexity of the energy. The N-ODE material model is able to capture synthetic data generated from closed-form material models, and it outperforms conventional models when tested against experimental data on skin, a highly nonlinear and anisotropic material. We also showcase the use of the N-ODE material model in finite element simulations of reconstructive surgery. The framework is general and can be used to model a large class of materials, especially biological soft tissues. We therefore expect our methodology to further enable data-driven methods in computational mechanics.

On Representing Strain Gradient Elastic Solutions of Boundary Value Problems by Encompassing the Classical Elastic Solution

The present work aims to primarily provide a general representation of the solution of the simplified elastostatics version of Mindlin’s Form II first-strain gradient elastic theory, which converges to the solution of the corresponding classical elastic boundary value problem as the intrinsic gradient parameters become zero. Through functional theory considerations, a solution representation of the one-intrinsic-parameter strain gradient elastostatic equation that comprises the classical elastic solution of the corresponding boundary value problem is rigorously provided for the first time. Next, that solution representation is employed to give an answer to contradictions arising by two well-known first-strain gradient elastic models proposed in the literature to describe the strain gradient elastostatic bending behavior of Bernoulli–Euler beams.

Global structure of positive solutions for semipositone nonlinear Euler-Bernoulli beam equation with Neumann boundary conditions

In this paper, we focus on the existence of positive solutions for nonlinear fourth-order Neumann boundary value problem where k 1 and k 2 are constants, λ > 0 is the bifurcation parameter, f ∈ C([0, 1] × ℝ+, ℝ), ℝ+ := [0, ∞). We first discuss the sign properties of Green’s function for the elastic beam boundary value problem, and then we show that there exists a global branch of solutions emanating from infinity under some different growth conditions. In addition, we prove that for λ near the bifurcation points, solutions of large norm are indeed positive. The technique for dealing with this paper relies on the global bifurcation theory.

On integral and differential formulations in nonlocal elasticity

The paper is concerned with comparative analysis of differential and integral formulations for boundary value problems in nonlocal elasticity. For the sake of simplicity, the focus is on an antiplane problem for a half-space with prescribed shear stress along the surface. In addition, 1D exponential kernel depending on the vertical coordinate is considered.First, a surface loading in the form of a travelling harmonic wave is studied. This provides a counter-example, revealing that within the framework of Eringen’s theory the solution to the differential model does not satisfy the equation of motion in nonlocal stresses underlying the related integral formulation.A more general differential setup, starting from singularly perturbed equations expressing the local stresses through the nonlocal ones, is also investigated. It is emphasised that the transformation of the original integral formulation to the differential one in question is only possible provided that two additional conditions on nonlocal stresses hold on the surface. As a result, the formulated problem subject to three boundary conditions appears to be ill-posed, in line with earlier observations for equilibrium of a nonlocal cantilever beam.Next, the asymptotic solution of the singularly perturbed problem, subject to a prescribed stress on the boundary, together with only one of the aforementioned extra conditions, is obtained at a small internal size. Such simplification may be justified when only one of the stresses demonstrates nonlocal behaviour; a similar assumption has been recently made within the so-called dilatational gradient elasticity. Three-term expansion is obtained, leading to a boundary value problem in local stresses over interior domain. The associated differential equations are identical to those proposed by Eringen, however, the derived effective boundary condition incorporates the effect of a nonlocal boundary layer which has previously been ignored. Moreover, the calculated nonlocal correction to the classical antiplane problem for an elastic half-space, coming from the boundary conditions is by order of magnitude greater than that appearing in the equations of motion. Finally, it is shown that the proposed effective condition supports an antiplane surface wave.

Frequency characteristics of built-in fiber-optic PEL-sensor to diagnose complex harmonic deformations within the polymer composite structure

Fundamentals of operation of an optical fiber piezoelectroluminescent (PEL) sensor inside a polymer composite structure at its cyclic loading are considered. The optical fiber PEL-sensor is considered as part of the composite/sensor electromechanical system taking into account the presence of anisotropy, piezoactivity and Maxwell-Wagner relaxation of the electric fields of the sensor elements. The purpose of the optical fiber PEL-sensor is to diagnose the inhomogeneous complex volumetric deformed state of a long cylindrical area (a neighborhood along the built-in linear sensor) inside a cyclically loaded composite structure. A numerical model has been developed to solve the 3D related boundary value problem of electric elasticity for a representative fragment of the system composite/sensor in the ANSYS package. The numerical modeling of deformation and electric harmonic fields inside the representative fragment was carried out; in particular, distributions of amplitudes of these fields in elements of the structure of the optical fiber PEL sensor were found. The resonant modes are revealed, and the analysis is given of regularities of frequency dependences for the real and imaginary parts of controlling and informative transfer coefficients of the built-in fiber-optic PEL-sensor in the composite/sensor system. Additionally, graphs of frequency dependencies of tangents of mechanical loss angles for various cases of deformation of the composite/sensor system are given. Damping of the composite/sensor system is carried out as a result of the conversion of some part of the mechanical energy (transmitted from the composite to the sensor during their joint deformation) into Joule heat by the fiber-optic PEL sensor with a subsequent dispersion. The latter is caused by the direct piezoelectric effect and Maxwell-Wagner relaxation of electric fields in the sensor elements. The frequency range of deformation of the composite/sensor system is set, in which the passive vibration damping mode is most effectively implemented. It is numerically confirmed that for the extreme high-frequency case of deformation of the composite/sensor system, relaxation processes are not implemented and, as a result, solutions for the controlling and informative transfer coefficients of the PEL-sensor practically coincide with previously obtained numerical solutions that did not take into account the electrical conductivity of the sensor structure elements.

EvoStencils: a grammar-based genetic programming approach for constructing efficient geometric multigrid methods

For many systems of linear equations that arise from the discretization of partial differential equations, the construction of an efficient multigrid solver is challenging. Here we present EvoStencils, a novel approach for optimizing geometric multigrid methods with grammar-guided genetic programming, a stochastic program optimization technique inspired by the principle of natural evolution. A multigrid solver is represented as a tree of mathematical expressions that we generate based on a formal grammar. The quality of each solver is evaluated in terms of convergence and compute performance by automatically generating an optimized implementation using code generation that is then executed on the target platform to measure all relevant performance metrics. Based on this, a multi-objective optimization is performed using a non-dominated sorting-based selection. To evaluate a large number of solvers in parallel, they are distributed to multiple compute nodes. We demonstrate the effectiveness of our implementation by constructing geometric multigrid solvers that are able to outperform hand-crafted methods for Poisson’s equation and a linear elastic boundary value problem with up to 16 million unknowns on multi-core processors with Ivy Bridge and Broadwell microarchitecture.

A guide to the finite and virtual element methods for elasticity

We present a systematic description and comparison of the Finite Element Method (FEM) with the relatively new Virtual Element Method (VEM) for solving boundary value problems in linear elasticity, including primal and mixed formulations. The description highlights the common base and the essential difference between FEM and VEM: discretisation of the same primal (Galerkin) and mixed weak formulations and assembly of element-wise quantities, but different approaches to element shape functions. The mathematical formulations are complemented with detailed description of the computer implementation of all methods, including all versions of VEM, which will benefit readers willing to develop their own computational framework. Numerical solutions of several boundary value problems are also presented in order to discuss the weaker and stronger sides of the methods.

Boundary influence on the fields and arrangement of the needle domains in barium titanate single crystals

The superposition theory is adopted to study the boundary value problem of needle domains in barium titanate single crystals, which decomposes into an infinite-medium solution and a complementary solution. The model of an equivalent edge dislocation and line charge is adopted to calculate the fields due to the discrete needle domains in an infinite medium. The complementary solutions are obtained from a linear elastic boundary value problem using the finite element method. The influence of boundaries on the fields and arrangement of the needles in thin barium titanate single crystals is analysed, by considering traction-free boundaries and using the plane stress approximation. According to the principle of minimum potential energy, it is found that pairs of parallel needle domains can be stabilized if the effective charge associated with the needle tips is reduced to 48% of the material’s full polarization charge. The evolution of comb-like arrays of needle domains under compressive mechanical load is simulated, and the results are found to be qualitatively similar to experimental observations which have not previously been explained theoretically.