Initial And Boundary Value Problem Research Articles

Global stability under dynamic boundary conditions of a nonlinear PDE model arising from reinforced random walks

This paper is devoted to the study of the global stability of classical solutions to an initial and boundary value problem (IBVP) of a nonlinear PDE system, converted from a model of reinforced random walks, subject to time-dependent boundary conditions. It is shown that under certain integrability conditions on the boundary data, classical solutions to the IBVP exist globally in time and the differences between the solutions and their corresponding ansatz, determined by the initial and boundary conditions, converge to zero, as time goes to infinity. Though the final states of the boundary functions are required to match, the boundary values do not necessarily equal to each other at any finite time. In addition, there is no smallness restriction on the magnitude of the initial perturbations. Furthermore, numerical simulations are performed to investigate the long-time dynamics of the IBVP when the final states of the boundary functions do not match or the boundary functions are periodic in time.

Estimation of Thermal Pollution Using Numerical Simulation of Energy Equation Coupled with Viscous Burgers’ Equation

In this paper, we implement energy equation coupled with viscous Burgers’ equation as a mathematical model for the estimation of thermal pollution of river water. The model is a nonlinear system of partial differential equations (PDEs) that read as an initial and boundary value problem (IBVP). For the numerical solution of the IBVP, we investigate an explicit second-order Lax- Wendroff type scheme for nonlinear parabolic PDEs. We present the numerical solutions graphically as a temperature profile, which shows good qualitative agreement with natural phenomena of heat transfer. We estimate the thermal pollution of water caused by industrialization on the bank of a river.

The Equal-Norm Multiple-Scale Trefftz Method for Solving the Nonlinear Sloshing Problem with Baffles

In this paper, the equal-norm multiple-scale Trefftz method combined with the implicit Lie-group scheme is applied to solve the two-dimensional nonlinear sloshing problem with baffles. When considering solving sloshing problems with baffles by using boundary integral methods, degenerate geometry and problems of numerical instability are inevitable. To avoid numerical instability, the multiple-scale characteristic lengths are introduced into T-complete basis functions to efficiently govern the high-order oscillation disturbance. Again, the numerical noise propagation at each time step is eliminated by the vector regularization method and the group-preserving scheme. A weighting factor of the group-preserving scheme is introduced into a linear system and then used in the initial and boundary value problems (IBVPs) at each time step. More importantly, the parameters of the algorithm, namely, the T-complete function, dissipation factor, and time step, can obtain a linear relationship. The boundary noise interference and energy conservation are successfully overcome, and the accuracy of the boundary value problem is also improved. Finally, benchmark cases are used to verify the correctness of the numerical algorithm. The numerical results show that this algorithm is efficient and stable for nonlinear two-dimensional sloshing problems with baffles.

Development and Implementation of Block Unification Multi-step Methods for the Solution of Second Order Ordinary Differential Equations

In this paper, linear multi-step hybrid block methods with three-, four- and five-step numbers are developed for approximating directly the solution of second order Initial and Boundary Value Problems (IBVPs). Multiple finite difference formulas are derived and combined in a block formulation to form a numerical integrator that provides direct solution to second order IBVPs over sub-intervals. A new class of orthogonal polynomials constructed as basis function to develop the hybrid block methods adopting collocation technique with a non-negative weight function. The scheme is applied as simultaneous integrator to second order initial value and boundary value problems of ODEs. The properties and convergence of the proposed method are discussed. The derived schemes were used to solve some problems and the numerical result shows the effectiveness, accuracy and superiority of the method over the existing methods found in the literature.

Numerical Solutions of Time Dependent Partial Differential Equations Using Weighted Residual Method With Piecewise Polynomials

We use Galerkin weighted residual (GWR) method to solve one dimensional heat and wave equations as initial and boundary value problems (IBVPs) numerically. Three special types of piecewise polynomials namely: Bernstein, Bernoulli and Legendre polynomials are used as basis functions to solve these IBVPs. A few examples are tested by the proposed method and then the results are compared with the solutions found in other existing methods. The numerical results obtained in this paper are in good agreement with the exact solutions.
 Dhaka Univ. J. Sci. 67(1): 5-12, 2019 (January)

Existence and uniqueness of a problem in thermo-elasto-plasticity with phase transitions in TRIP steels under mixed boundary conditions

Abstract In this paper a complex model describing thermo-elasto-plasticity, phase transitions (PT) and transformation-induced plasticity (TRIP) is studied. The main objective is the analysis of the corresponding initial and boundary value problem (IBVP) considering linearized thermo-elastic dissipation and a viscosity-like regularization.

Ductile damage prediction in sheet and bulk metal forming

Abstract This paper is dedicated to the presentation of an advanced 3D numerical methodology for virtual sheet and/or bulk metal forming simulation to predict the anisotropic ductile defects occurrence. First, the detailed formulation of thermodynamically-consistent fully coupled and fully anisotropic constitutive equations is given. The proposed constitutive equations account for the main material nonlinearities as the anisotropic plastic flow, the mixed isotropic and kinematic hardening and the anisotropic ductile damage under large inelastic strains. Second, the related numerical aspects required to solve the initial and boundary value problem (IBVP) are very briefly presented in the framework of the 3D finite element method. The global resolution schemes as well as the local integration schemes of the fully coupled constitutive equations are briefly discussed. Finally, some typical examples of sheet and bulk metal forming processes are numerically simulated using the proposed numerical methodology.

Well-posedness of the two-dimensional generalized Benjamin-Bona-Mahony equation on the upper half plane

This paper focuses on the two-dimensional Benjamin-Bona-Mahony and Benjamin-Bona-Mahony-Burgers equations with a general flux function. The aim is at the global (in time) well-posedness of the initial-and boundary-value problem for these equations defined in the upper half-plane. Under suitable growth conditions on the flux function, we are able to establish the global well-posedness in a Sobolev class. When the initial- and boundary-data become more regular, the corresponding solutions are shown to be classical. In addition, the continuous dependence on the data is also obtained.

Effects of Generalized Fundamental Solution (GFS) on Generalized Integral Representation Method (GIRM)

Integral Representation Method (IRM) is one of convenient methods to solve Initial and Boundary Value Problems (IBVP). It can be applied to irregular mesh, and the solution is stable and accurate. IRM is developed to Generalized Integral Representation Method (GIRM) to treat any kinds of problems including nonlinear problems. In GIRM, Generalized Fundamental Solution (GFS) is used instead of Fundamental Solution (FS) in IRM. Since GFS is not limited to one, the effects of individual GFSs must be clarified. The continuity of GFS is related to the characteristics of individual GFSs.

From Integral Representation Method (IRM) to Generalized Integral Representation Method (GIRM)

Integral Representation Method (IRM) is one of convenient methods to solve Initial and Boundary Value Problems (IBVP). It can be applied to irregular mesh, and the solution is stable and accurate. However, it was originally developed for linear equations with known fundamental solutions. In order to apply to general nonlinear equations, we must generalize the method. In the present paper, a generalization of IRM (GIRM) is discussed and applied to specific problems and the numerical solutions obtained. The numerical results are stable and accurate. The generalized method is called Generalized Integral Representation Method (GIRM). Brief explanations on the relationships with other numerical methods are also given.

Application of the Generalized Integral Representation Method (GIRM) to Tidal Wave Propagation

Integral Representation Method (IRM) is one of convenient methods to solve Initial and Boundary Value Problems (IBVP). It can be applied to irregular mesh, and the solution is stable and accurate. IRM is developed to Generalized Integral Representation Method (GIRM) to treat any kinds of problems including nonlinear problems. In GIRM, Generalized Fundamental Solution (GFS) is used instead of Fundamental Solution (FS) in IRM. We can use a variety of GFSs in GIRM. The effects of typical GFSs are investigated. In the present paper, an application of GIRM to tidal wave propagation is discussed, and the time evolution involves the second order time derivatives. An explicit time evolution is used successfully in the present paper.

Existence of solutions for linear hyperbolic initial-boundary value problems in a rectangle

In a recent article, we achieved the well-posedness of linear hyperbolic initial and boundary value problems (IBVP) in a rectangle via semigroup method, and we found that there are only two elementary modes called hyperbolic and elliptic modes in the system. It seems that, there is only one set of boundary conditions for the hyperbolic mode, while there are infinitely many sets of boundary conditions for the elliptic mode, which can lead to well-posedness. In this article, we continue to consider linear hyperbolic IBVP in a rectangle in the constant coefficients case and we show that there are also infinitely many sets of boundary conditions for hyperbolic mode which will lead to the existence of a solution. We also have uniqueness in some special cases. The boundary conditions satisfy the reflection conditions introduced in Section 3, which turn out to be equivalent to the strictly dissipative conditions.

The linear hyperbolic initial and boundary value problems in a domain with corners

In this article, we consider linear hyperbolic Initial and Boundary Value Problems (IBVP) in a rectangle (or possibly curvilinear polygonal domains) in both the constant and variable coefficients cases. We use semigroup method instead of Fourier analysis to achieve the well-posedness of the linear hyperbolic system, and we find by diagonalization that there are only two elementary modes in the system which we call hyperbolic and elliptic modes. The hyperbolic system in consideration is either symmetric or Friedrichs-symmetrizable.

Micromorphic approach for finite gradient-elastoplasticity fully coupled with ductile damage: Formulation and computational aspects

It is well established that the use of inelastic constitutive equations accounting for induced softening, leads to pathological space (mesh) and time discretization dependency of the numerical solution of the associated Initial and Boundary Value Problem (IBVP). To avoid this drawback, many less or more approximate solutions have been proposed in the literature in order to regularize the IBVP and to obtain numerical solutions which are, at convergence, much less sensitive to the space and the time discretization. The basic idea behind these regularization techniques is the formulation of nonlocal constitutive equations by introducing some effects of characteristic lengths representing the materials microstructure. In this work, using the framework of generalized nonlocal continua, a thermodynamically-consistent micromorphic formulation using appropriate micromorphic state variables and their first gradients, is proposed in order to extend the classical local constitutive equations by incorporating appropriate characteristic internal lengths. The isotropic damage, the isotropic and the kinematic hardenings are supposed to carry the targeted micromorphic effects. First the theoretical aspects of this fully coupled micromorphic formulation is presented in details and the proposed generalized balance equations as well as the fully coupled micromorphic constitutive equations deduced. The associated numerical aspects in the framework of the classical Galerkin-based FE formulation are briefly discussed in the special case of micromorphic damage. Specifically, the formulation of 2D finite elements with additional degrees of freedom (d.o.f.), the dynamic explicit global resolution scheme as well as the local integration scheme, to compute the stress tensor and the state variables at each integration point of each element, are presented. Application is made to the typical uniaxial tension specimen under plane strain conditions in order to chow the predictive capabilities of the proposed micromorphic model, particularly against its ability to give (at convergence) a mesh independent solution even for high values of the ductile damage (i.e., the macroscopic cracks).

A simple damage-gradient enhanced elastoplastic formulation and its numerical implementation

This paper presents a ductile damage-gradient based nonlocal and fully coupled elastoplastic constitutive equations by adding a Helmholtz equation to regularize the initial and boundary value problem (IBVP) exhibiting some damage induced softening. First, a thermodynamically-consistent formulation of gradient-regularized plasticity fully coupled with isotropic ductile damage and accounting for mixed non linear isotropic and kinematic hardening is presented. For the sake of simplicity, only a simplified version of this model based on von Mises isotropic yield function and accounting for the single nonlinear isotropic hardening is studied and implemented numerically using an in house FE code. An additional partial differential equation governing the evolution of the nonlocal isotropic damage is added to the equilibrium equations and the associated weak forms derived to define the IBVP (initial and boundary value problem). After the time and space discretization, two algebraic equations: one highly nonlinear associated with the equilibrium equation and the second purely linear associated with the damage non locality equation are obtained. Over a typical load increment, the first equation is solved iteratively thanks to the Newton-Raphson scheme and the second equation is solved directly to compute the nonlocal damage \hbox{} D at each node. All the constitutive equations are “strongly” affected by this nonlocal damage variable transferred to each integration point. Some applications show the ability of the proposed approach to obtain a mesh independent solution for a fixed value of the length scale parameter. Comparisons between fully local and nonlocal solutions are given.

The stability of an isothermal gaseous sphere with gravitation

A dispersion relation has been obtained for waves propagating in an inhomogeneous gas with viscosity and gravity in the direction of the density gradient. It has been shown that a stationary distribution of isothermal gas is unstable under gravitational—convective perturbations. It is suggested that the initialand boundary-value problems of the non-stationary dynamics of an isothermal gas with gravity but without viscosity are incorrect and require regularization, for example, by adding viscosity. The presence of viscosity does not prevent the development of the Jeans long-wave instability. Numerical experiments based on a fully three-dimensional, non-stationary numerical code have demonstrated the development of a collapse inside an isothermal cloud.

Non-linear analysis of the consolidation of an elastic saturated soil with incompressible fluid and variable permeability by FEM

Research interest in the mechanical behaviour of soils is growing as a result of an increasing number of geomechanical problems involving consolidation effects. The main aim of this paper is to validate and to solve a model for consolidation of an elastic saturated soil with incompressible fluid and variable permeability. Firstly, we prove the existence and uniqueness of the solution of the variational problem corresponding to an initial and boundary value problem (IBVP): a special case of the Biot’s ‘consolidation of clay’ model (where the applied forces depend on time). Secondly, we prove the convergence of the method using a technique based on the proof of solution’s existence. Finally, we then solved this constitutive model by the finite element method (FEM) employing repeated fixed point techniques in order to obtain the results for displacement and pore water pressure. The pore fluid is considered incompressible. The results of the numerical experiments are compared with analytical solutions and, in cases where such solutions do not exist, with experimental data. Therefore, the model can be used for quantitative predictions of consolidation behaviour of soils with permeability dependent on the settlement.

Isotropic Elastoplasticity Fully Coupled with Non-Local Damage

This paper presents a simple damage-gradient based elastoplastic model with non linear isotropic hardening in order to regularize the associated initial and boundary value problem (IBVP). Using the total energy equivalence hypothesis, fully coupled constitutive equations are used to describe the non local damage induced softening leading to a mesh independent solution. An additional partial differential equation governing the evolution of the non local isotropic damage is added to the classical equilibrium equations and associated weak forms derived. This leads to discretized IBVP governed by two algebric systems. The first one, associated with equilibrium equations, is highly non linear and can be solved by an iterative Newton Raphson method. The second one, related to the non local damage, is a linear algebric system and can be solved directly to compute the non local damage variable at each load increment. Two fields, linear interpolation triangular element with additional degree of freedom is terms of the non local damage variable is constructed. The non local damage variable is then transferred from mesh nodes to the quadrature (or Gauss) points to affect strongly the elastoplastic behavior. Two simple 2D examples are worked out in order to investigate the ability of proposed approach to deliver a mesh independent solution in the softening stage.

Mathematical modelling and study of the consolidation of an elastic saturated soil with an incompressible fluid by FEM

The main aim of this paper is to validate and to solve a model for consolidation of an elastic saturated soil with incompressible fluid. Firstly, we prove the existence and uniqueness of the solution of the variational problem corresponding to an initial and boundary value problem (IBVP): a special case of the Biot’s ‘consolidation of clay’ model (where the applied forces depend on time). Secondly, we prove the stability of the method as well as the estimation of the error by using semi-discretization in time. Finally, we then solved this one by the finite element method (FEM) employing repeated fixed point techniques in order to obtain the results for displacement and pore water pressure. The pore fluid is considered incompressible. The results of the numerical experiments are compared with analytical solutions and, in cases where such solutions do not exist, with experimental data.

Existence of a solution and averaging of nonlinear multifrequency problems with delay

The method of averaging over the fast variables is applied to the investigation of multifrequency systems with linearly transformed argument. We prove the existence of solutions of the initial-and boundary-value problems in a small neighborhood of the solution of the averaged problem and estimate the error of the method of averaging for slow variables.