Non-regular Domains Research Articles

A multimesh finite element method for integral nonlocal elasticity using mesh-decoupling technique

This study presents a generalized multimesh nonlocal finite element method (M2-FEM) that addresses several long-standing challenges in the numerical simulation of Eringen’s integral elasticity theories, often used to model nonlocal effects. The major challenges in the numerical simulation of integral boundary value problems are primarily rooted in the coupling of the spatial discretization of the global (parent) and integral (child) domains, which severely restricts both the applicability and the computational efficiency of existing algorithms by imposing an implicit trade-off between the accuracy of the child domain and the resources required by the overall parent domain. One of the defining contributions of this study consists in the development of a mesh-decoupling technique that generates isolated sets of meshes such that the parent and child domains can be discretized and approximated independently. This mesh-decoupling has multi-fold advantages on the simulation of integral theories such that, when compared to existing state-of-the-art techniques, the proposed algorithm achieves a greater ability to adopt generalized integral kernel functions, the ability to handle non-regular (non-rectangular) domains via unstructured meshing, and simultaneously better numerical accuracy and efficiency (hence allowing greater flexibility in both mesh size and computational cost trade-off decisions). Both benchmark studies and several case studies are conducted to assess the effectiveness of M2-FEM. Simulation results confirm that M2-FEM can accurately simulate integral nonlocal problems and demonstrate its multi-fold computational advantages over existing mesh-coupling-based nonlocal finite element methods. Overall, the proposed M2-FEM algorithm is very general and it can be applied to a variety of integral theories, even beyond nonlocal elasticity.

Numerical investigation for a boundary optimal control of reaction-advection-diffusion equation using penalization technique

<p>The objective of this paper is to describe the problem of boundary optimal control of the reaction-advection-diffusion equation for not very regular Dirichlet data and enumerate its qualitative properties. We check that the state equation is well-posed and we introduce the penalization technique to take into account the boundary condition of the Dirichlet type. Then, we consider the corresponding optimal boundary control problem and give the optimality conditions. Finally, we conducted a numerical investigation of the convergence of the solution of the penalized problem to the solution of the non-penalized one when the penalty parameter tends to zero in regular and non-regular domains.</p>

Stabilization of Coupled Wave Equations with Viscous Damping on Cylindrical and Non-regular Domains: Cases Without the Geometric Control Condition

In this paper, we investigate the direct and indirect stability of locally coupled wave equations with local viscous damping on cylindrical and non-regular domains without any geometric control condition. If only one equation is damped, we prove that the energy of our system decays polynomially with the rate $$t^{-\frac{1}{2}}$$ if the two waves have the same speed of propagation, and with rate $$t^{-\frac{1}{3}}$$ if the two waves do not propagate at the same speed. Otherwise, in case of two damped equations, we prove a polynomial energy decay rate of order $$t^{-1}$$ .

On the resolution of the heat equation in unbounded non-regular domains of R³

We will prove well posedness and regularity results for the bidimensional heat equation, subject to mixed Dirichlet-Neumann type boundary conditions on the parabolic boundary of an unbounded (in one space variable direction) time-dependent domain. Our results are proved in anisotropic Hilbertian Sobolev spaces by using the domain decomposition method. This work complements the results obtained in [13] in the one-space variable case.

All self-adjoint extensions of the magnetic Laplacian in nonsmooth domains and gauge transformations

We use boundary triples to find a parametrization of all self-adjoint extensions of the magnetic Schrodinger operator, in a quasi-convex domain~$\Omega$ with compact boundary, and magnetic potentials with components in $\textrm{W}^{1}_{\infty}(\overline{\Omega})$. This gives also a new characterization of all self-adjoint extensions of the Laplacian in nonregular domains. Then we discuss gauge transformations for such self-adjoint extensions and generalize a characterization of the gauge equivalence of the Dirichlet magnetic operator for the Dirichlet Laplacian; the relation to the Aharonov-Bohm effect, including irregular solenoids, is also discussed. In particular, in case of (bounded) quasi-convex domains it is shown that if some extension is unitarily equivalent (through the multiplication by a smooth unit function) to a realization with zero magnetic potential, then the same occurs for all self-adjoint realizations.

Growth and concentration dependences of properties of LiNbO3:Tb crystals grown in a single technological cycle

A series of crystals LiNbO3:Tb ([Tb] = 0.1 ÷ 2.89 wt% in a crystal) and nominally pure near-stoichiometric crystal (NSLN) has been grown in a single technological cycle. Crystallization processes have been studied along with concentration dependences of physical, chemical and structural characteristics of LiNbO3:Tb crystals. LiNbO3:Tb atomic structure models have been analyzed by full-profile analysis of XRD patterns of powders. Regular and non-regular domain growth microstructures together with periodic nano-sized structures have been determined in as-grown LiNbO3:Tb crystals using optical and atomic-force microscopy.

An effective pure meshfree method for 1D/2D time fractional convection-diffusion problems on irregular geometry

An easy implemented and effective pure meshfree method is first developed to solve the 1D/2D constant/variable-order time fractional convection-diffusion equation (TF-CDE) on non-regular domain with two boundary conditions in this paper. The proposed method (CSPH-FDM) is derived from that the finite difference scheme (FDM) for Caputo time fractional derivative and a corrected smoothed particle hydrodynamics (CSPH) without kernel derivative for spatial derivatives. In the proposed CSPH-FDM, the high-order spatial derivative is divided into multi first-order derivatives and solved continuously by the CSPH, the Neumann boundary condition can be accurately treated, the two distribution cases of the local refinement and irregular particles or the arbitrary irregular shape domain can be easily and effectively implemented by the CSPH. To demonstrate the validity and numerical convergent order of the proposed method, several 1D/2D analytical examples with local refinement and irregular particles distributions or on complex geometries are first investigated, in which a four-order derivate problem with Neumann boundary is also considered. Subsequently, the CSPH-FDM is extended to predict the solute moving process versus time by two TF-CDEs on different irregular domains and compared with other numerical results. All the numerical results show the flexible application ability and reliability of the present method.

The behavior of solutions of nonlinear degenerate parabolic equations in nonregular domains and removability of singularity on boundary

The behavior of solutions of nonlinear degenerate parabolic equations in nonregular domains and removability of singularity on boundary

A novel DEM approach for modeling brittle elastic media based on distinct lattice spring model

The Discrete Element Method (DEM), also known as Distinct Element Method (DEM), is extensively used to study divided media such as granular materials. When brittle failure occurs in continuum such as concrete or ceramics, the considered media can be viewed as divided. In such cases, DEM offers an interesting way to study and simulate complex fracture phenomena such as crack branching, crack extension, crack deviation under coupled mode or crack lip closure with friction.The fundamental difficulty with DEM is the inability of the method to deal directly with the constitutive equations of continuum mechanics. DEM uses force–displacement interaction laws between particles instead of stress–strain relationships. Generally, this difficulty is bypassed by using inverse methods, also known as calibration processes, able to translate macroscopic stress–strain relationships into local force–displacement interaction laws compatible within DEM frameworks. However, this calibration process may be fastidious and really hard to manage.The presented work proposes to improve the Distinct Lattice Spring Model in order to deal with non-regular domains, by using Voronoi cells, which allow to completely fill the volume space of discrete domains. With this approach, the rotational effects must be included in the contact formulation, which enables the management of large rigid body rotations. This work also introduces a simple method to manage brittle fracture. Using non-regular domains avoids the cracks paths conditioning, and allows to reproduce quantitatively the Brazilian test, very popular in the rock mechanics community.

On a Second Order Linear Parabolic Equation with Variable Coefficients in a Non-Regular Domain of R³

On a Second Order Linear Parabolic Equation with Variable Coefficients in a Non-Regular Domain of R³

Circulant embedding with QMC: analysis for elliptic PDE with lognormal coefficients

In a previous paper (Graham et al. in J Comput Phys 230:3668–3694, 2011), the authors proposed a new practical method for computing expected values of functionals of solutions for certain classes of elliptic partial differential equations with random coefficients. This method was based on combining quasi-Monte Carlo (QMC) methods for computing the expected values with circulant embedding methods for sampling the random field on a regular grid. It was found capable of handling fluid flow problems in random heterogeneous media with high stochastic dimension, but no convergence theory was provided. This paper provides a convergence analysis for the method in the case when the QMC method is a specially designed randomly shifted lattice rule. The convergence result depends on the eigenvalues of the underlying nested block circulant matrix and can be independent of the number of stochastic variables under certain assumptions. In fact the QMC analysis applies to general factorisations of the covariance matrix to sample the random field. The error analysis for the underlying fully discrete finite element method allows for locally refined meshes (via interpolation from a regular sampling grid of the random field). Numerical results on a non-regular domain with corner singularities in two spatial dimensions and on a regular domain in three spatial dimensions are included.

Scattered data fitting of Hermite type by a weighted meshless method

In many practical problems, it is often desirable to interpolate not only the function values but also the values of derivatives up to certain order, as in the Hermite interpolation. The Hermite interpolation method by radial basis functions is used widely for solving scattered Hermite data approximation problems. However, sometimes it makes more sense to approximate the solution by a least squares fit. This is particularly true when the data are contaminated with noise. In this paper, a weighted meshless method is presented to solve least squares problems with noise. The weighted meshless method by Gaussian radial basis functions is proposed to fit scattered Hermite data with noise in certain local regions of the problem’s domain. Existence and uniqueness of the solution is proved. This approach has one parameter which can adjust the accuracy according to the size of the noise. Another advantage of the weighted meshless method is that it can be used for problems in high dimensions with nonregular domains. The numerical experiments show that our weighted meshless method has better performance than the traditional least squares method in the case of noisy Hermite data.

Another note on the embedding of the Sobolev space for the limiting exponent

The embedding of the Sobolev spaces W p s (ℝ n ) in a Lizorkin-type space of locally summable functions of zero smoothness is established. This result is extended to the case of the embedding of Sobolev spaces on nonregular domains of n-dimensional Euclidean space. The formulation of the theorem depends on the geometric parameters of the domain of the functions.

Composite mesh generator for CFD problems

In present paper a brief introduction of HybMesh grid generator which uses composite approach is given. The process of complicated area meshing using HybMesh generator consists of sequential building structured prototype grids in relatively simple geometry, mapping them to a non-regular domains and superposing to assemble resulting grid. Transitional areas between two superposed low level grids are filled with triangular cells. Currently only 2D algorithms of such approach are implemented; 3D grids can only be restored as a result of extrusion or revolution of 2D objects.

Analysis of the Time Fractional 2-D Diffusion-Wave Equation via Moving Least Square (MLS) Approximation

In this paper, the time fractional two-dimensional diffusion-wave equation defined by Caputo sense for ( $$1<\alpha <2$$ ) is analyzed by an efficient and accurate computational method namely meshless local Petrov–Galerkin (MLPG) method which is based on the Galerkin weak form and moving least squares (MLS) approximation. We consider a general domain with Dirichlet boundary conditions further to given initial values as continuous functions. Meshless Galerkin weak form is adopted to the interior nodes while the meshless collocation technique is applied for the nodes on the boundaries of the domain. Since Dirichlet boundary condition is imposed directly therefore the general domains are also applicable easily. In MLPG method, the MLS approximation is usually used to construct shape functions which plays important rule in the convergence and stability of the method. It is proved the method is unconditionally stable in some sense. Two numerical examples are presented, one of them with the regular domain and the other one with non-regular domain, and satisfactory agreements are achieved.

Robust fitting of Zernike polynomials to noisy point clouds defined over connected domains of arbitrary shape.

A new method for fitting a series of Zernike polynomials to point clouds defined over connected domains of arbitrary shape defined within the unit circle is presented in this work. The method is based on the application of machine learning fitting techniques by constructing an extended training set in order to ensure the smooth variation of local curvature over the whole domain. Therefore this technique is best suited for fitting points corresponding to ophthalmic lenses surfaces, particularly progressive power ones, in non-regular domains. We have tested our method by fitting numerical and real surfaces reaching an accuracy of 1 micron in elevation and 0.1 D in local curvature in agreement with the customary tolerances in the ophthalmic manufacturing industry.

Dirichlet Problem for Complex Poisson Equation in a Half Hexagon Domain

The parqueting-reflection method is applied to a nonregular domain and the harmonic Green function for the half hexagon is constructed. The related Dirichlet problem for the Poisson equation is solved explicitly.

The numerical simulation of the phase field crystal (PFC) and modified phase field crystal (MPFC) models via global and local meshless methods

Abstract In the current work, we consider the two-dimensional time-dependent phase field crystal (PFC) and the modified phase field crystal models (MPFC) to obtain their numerical solutions. For this purpose, we apply two numerical meshless methods based on radial basis functions (RBFs) and also two meshless methods which are based on moving least squares method (MLS). Four techniques developed in this paper are: globally radial basis functions (GRBFs), radial basis functions pseudo-spectral (RBFs-PS), moving least squares (MLS) and generalized moving least squares (GMLS) approximations. Two methods based on RBFs are global and the other methods based on moving least squares are local procedures. As is well-known, the meshless methods are suitable techniques for the numerical solution of partial differential equations on regular and non-regular domains with different choices of grids in high-dimensions. Applying the new methods on spatial domain and also using the semi-implicit scheme for time variable, yield a linear system of algebraic equations. To solve this linear system, the LU decomposition algorithm and command “backslash” in MATLAB software (for GMLS method) are applied. The implementation of boundary conditions in the RBFs-PS is applied directly, because the boundary conditions of the mentioned problems are periodic. Some numerical results show that the obtained simulations via four proposed methods are acceptable for approximating the solution of models investigated in the current paper. Moreover in the Appendix of the paper, a MATLAB code for GMLS method is written.

Parabolic equations with Cauchy–Dirichlet boundary conditions in a non-regular domain of ℝN+1

Abstract. New results on the existence, uniqueness and maximal regularity of a solution are given for a parabolic equation set in a non-regular domain Q = ( t , x 1 ) ∈ ℝ 2 : 0 &lt; t &lt; T ; ϕ 1 ( t ) &lt; x 1 &lt; ϕ 2 ( t ) × ∏ i = 1 N - 1 ] 0 , b i [ $ Q=\bigl \lbrace (t,x_1) \in \mathbb {R}^{2}:0&amp;lt;t&amp;lt;T;\; \varphi _1(t) &amp;lt;x_1&amp;lt;\varphi _{2}(t) \bigr \rbrace \times \prod _{i=1}^{N-1}{]0,b_i[} $ of ℝN+1, with Cauchy–Dirichlet boundary conditions, under some assumptions on the functions ϕ 1 $\varphi _1$ and ϕ 2 $\varphi _2$ . The right-hand side term of the equation is taken in L 2 ( Q ) $L^{2}(Q)$ . The method used is based on the approximation of the domain Q by a sequence of sub-domains ( Q n ) $(Q_n)$ which can be transformed into regular domains. This work is an extension of the two space variables case studied in [Differ. Equ. Appl. 2 (2010), no. 2, 251–263].

Existence and uniqueness results for parabolic equations with Robin type boundary conditions in a non-regular domain of [formula omitted

This paper is devoted to the analysis of the following two-space dimensional linear parabolic equation ∂tu-∂x12u-∂x22u=f, subject to Robin type conditions ∂x1u+βu=0, on the lateral boundary, where the coefficient β satisfies suitable non-degeneracy assumptions. The right-hand side f of the equation is taken in L2. The problem is set in a domain of the form t,x1∈R2:0<t<T,φ1(t)<x1<φ2(t)×]0,b[, where φ1 and φ2 are smooth functions. One of the main issues of this work is that the domain can possibly be non-regular, for instance, the singular case where φ1(0)=φ2(0) is allowed. For this purpose, the domain decomposition method is employed.