Localized direct boundary-domain integro-differential formulations for incremental elasto-plasticity of inhomogeneous body

A quasi-static mixed boundary value problem of incremental elasto-plasticity for a continuously inhomogeneous body is considered. Using the two-operator Green–Betti formula and the fundamental solution of a reference homogeneous linear elasticity problem, with frozen initial or tangent elastic coefficients, a boundary-domain integro-differential formulation of the elasto-plastic problem is presented, with respect to the displacement rates and their gradients. Using a cut-off function approach, the corresponding localized parametrix of the reference problem is constructed to reduce the elasto-plastic problem to a nonlinear localized boundary-domain integro-differential equation. Algorithms of mesh-based and mesh-less discretizations are presented resulting in sparsely populated systems of nonlinear algebraic equations for the displacement increments.     

Radial integration boundary integral and integro-differential equation methods for two-dimensional heat conduction problems with variable coefficients

This paper presents new formulations of the radial integration boundary integral equation (RIBIE) and the radial integration boundary integro-differential equation (RIBIDE) methods for the numerical solution of two-dimensional heat conduction problems with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the boundary-value problem (BVP) to a boundary-domain integral equation (BDIE) or boundary-domain integro-differential equation (BDIDE). The radial integration method is then employed to convert the domain integrals arising in both BDIE and BDIDE methods into equivalent boundary integrals. The resulting formulations lead to pure boundary integral and integro-differential equations with no domain integrals. Numerical examples are presented for several simple problems, for which exact solutions are available, to demonstrate the efficiency of the proposed methods.     

Mesh-based numerical implementation of the localized boundary-domain integral-equation method to a variable-coefficient Neumann problem

An implementation of the localized boundary-domain integral-equation (LBDIE) method for the numerical solution of the Neumann boundary-value problem for a second-order linear elliptic PDE with variable coefficient is discussed. The LBDIE method uses a specially constructed localized parametrix (Levi function) to reduce the BVP to a LBDIE. After employing a mesh-based discretization, the integral equation is reduced to a sparse system of linear algebraic equations that is solved numerically. Since the Neumann BVP is not unconditionally and uniquely solvable, neither is the LBDIE. Numerical implementation of the finite-dimensional perturbation approach that reduces the integral equation to an unconditionally and uniquely solvable equation, is also discussed.     

Numerical solution of two-dimensional mixed problems with variable coefficients by the boundary-domain integral and integro-differential equation methods

This paper presents a formulation of the boundary-domain integral equation (BDIE) and the boundary-domain integro-differential equation (BDIDE) methods for the numerical solution of two-dimensional mixed boundary-value problems (BVP) for a second-order linear elliptic partial differential equation (PDE) with variable coefficients. The methods use a specially constructed parametrix (Levi function) to reduce the BVP to a BDIE or BDIDE. The numerical formulation of the BDIDE employs an approximation for the boundary fluxes in terms of the potential function within the domain cells; therefore, the solution is fully described in terms of the variation of the potential function along the boundary and domain. Linear basis functions localised on triangular elements and standard quadrature rules are used for the calculation of boundary integrals. For the domain integrals, we have implemented Gaussian quadrature rules for two dimensions with Duffy transformation, by mapping the triangles into squares and eliminating the weak singularity in the process. Numerical examples are presented for several simple problems with square and circular domains, for which exact solutions are available. It is shown that the present method produces accurate results even with coarse meshes. The numerical results also show that high rates of convergence are obtained with mesh refinement.     

Localized direct boundary-domain integro-differential formulations for scalar nonlinear boundary-value problems with variable coefficients

Mixed boundary-value Problems (BVPs) for a second-order quasi-linear elliptic partial differential equation with variable coefficients dependent on the unknown solution and its gradient are considered. Localized parametrices of auxiliary linear partial differential equations along with different combinations of the Green identities for the original and auxiliary equations are used to reduce the BVPs to direct or two-operator direct quasi-linear localized boundary-domain integro-differential equations (LBDIDEs). Different parametrix localizations are discussed, and the corresponding nonlinear LBDIDEs are presented. Mesh-based and mesh-less algorithms for the LBDIDE discretization are described that reduce the LBDIDEs to sparse systems of quasi-linear algebraic equations.     

Numerics of boundary-domain integral and integro-differential equations for BVP with variable coefficient in 3D

A numerical implementation of the direct boundary-domain integral and integro-differential equations, BDIDEs for treatment of the Dirichlet problem for a scalar elliptic PDE with variable coefficient in a three-dimensional domain is discussed. The mesh-based discretisation of the BDIEs with tetrahedron domain elements in conjunction with collocation method leads to a system of linear algebraic equations (discretised BDIE). The involved fully populated matrices are approximated by means of the H-Matrix/adaptive cross approximation technique. Convergence of the method is investigated.     

A numerical solution of nonlinear parabolic-type Volterra partial integro-differential equations using radial basis functions

In this paper, an effective numerical method for solving nonlinear Volterra partial integro-differential equations is proposed. These equations include the partial differentiations of an unknown function and the integral term containing the unknown function which is the “memory” of problem. This method is based on radial basis functions (RBFs) and finite difference method (FDM) which provide the approximate solution. These techniques play the important role to reduce a nonlinear partial integro-differential equation to a linear system of equations. Some illustrative examples are shown to describe the method. Numerical examples confirm the validity and efficiency of the presented method.     

Banach空间非线性混合型微分-积分方程解的存在唯一性

在较宽的条件下研究了Banach空间中非线性混合型微分—积分初值问题解的存在唯一性及解的迭代逼近和误差估计，改进并推广了最近的一些结果。     

非线性分数阶积分微分方程边值问题正解的存在性

研究一类具有两个分数阶导数项的非线性分数阶积分微分方程积分边值问题。首先将原问题转化为只有一个导数项的等价形式,通过定义等价问题的上下解,再利用单调迭代技术建立了原问题正解的存在性与唯一性定理,给出了求其唯一正解的迭代格式和误差估计。最后给出实例说明所得结论的有效性和适用性。     

A mathematical model of a biosensor

This paper is concerned with the modelling of the evolution of a chemical reaction within a small cell. Mathematically the problem consists of a heat equation with nonlinear boundary conditions. Through an integro-differential equation reformulation, an asymptotic result is derived, a perturbation solution is developed, and a modified product integration method is discussed. Finally, an alternative integral formulation is presented which acts as a check on the previous results and permits high accuracy numerical solutions.     

Closed-form approximation and numerical validation of the influence of van der Waals force on electrostatic cantilevers at nano-scale separations

In this paper the two-point boundary value problem (BVP) of the cantilever deflection at nano-scale separations subjected to van der Waals and electrostatic forces is investigated using analytical and numerical methods to obtain the instability point of the beam. In the analytical treatment of the BVP, the nonlinear differential equation of the model is transformed into the integral form by using the Green's function of the cantilever beam. Then, closed-form solutions are obtained by assuming an appropriate shape function for the beam deflection to evaluate the integrals. In the numerical method, the BVP is solved with the MATLAB BVP solver, which implements a collocation method for obtaining the solution of the BVP. The large deformation theory is applied in numerical simulations to study the effect of the finite kinematics on the pull-in parameters of cantilevers. The centerline of the beam under the effect of electrostatic and van der Waals forces at small deflections and at the point of instability is obtained numerically. In computing the centerline of the beam, the axial displacement due to the transverse deformation of the beam is taken into account, using the inextensibility condition. The pull-in parameters of the beam are computed analytically and numerically under the effects of electrostatic and/or van der Waals forces. The detachment length and the minimum initial gap of freestanding cantilevers, which are the basic design parameters, are determined. The results of the analytical study are compared with the numerical solutions of the BVP. The proposed methods are validated by the results published in the literature.     

Application of the boundary-domain element method to the pre/post-buckling problem of von Kármán plates

Based on the BEM formulations for the finite deflection problem of von-Kármán-type plates, this paper presents an incremental boundary-domain element method for the pre/post-buckling problem of thin elastic plates. As the governing equations involve the coupled in-plane and out-of-plane deformations as the nonlinear terms, the boundary integral equations are formulated in terms of the increment by using the fundamental solutions for the linear parts of the differential operators. Some of the innovations are made in order to improve the accuracy and accelerate the convergence of the solution procedure. The load-incrementation method and also the arc-length-incrementation method are employed for each incremental step. Numerical analysis is carried out and the results are compared with the available analytical solutions to demonstrate the effectiveness of the proposed method.     

Nonlinear dynamics of interface between dielectric liquids in vertical electric and gravity fields

The development of instability on the interface between dielectric liquids in vertical electric and gravity fields has been studied. The possibility of a special regime of motion is established, in which the velocity potential is linearly related to the electric field potential. An integro-differential equation is derived for this regime, which describes a weakly nonlinear evolution of the interface. This equation admits the existence of broad classes of exact solutions that determine the dynamics of both periodic and localized perturbations of the interface.     

Deformation and fracture of materials near spheroidal inclusions

We have proposed a model of the deformation and fracture of an elastoplastic body with an elastic spheroidal inclusion. The problem has been reduced to the solution of an integro-differential equation, and its numeral solution has been obtained by the method of mechanical quadratures. Using the strain criterion of crack initiation in the neighborhood of an inclusion, we have established the main parameters affecting local fracture. The results of investigations are presented in the form of plots.     

Mesoscopic aspects of the computational homogenization with meshless modeling for masonry material

The multiscale homogenization scheme is becoming a diffused tool for the analysis of heterogeneous materials as masonry since it allows dealing with the complexity of formulating closed-form constitutive laws by retrieving the material response from the solution of a unit cell (UC) boundary value problem (BVP). The robustness of multiscale simulations depends on the robustness of the nested macroscopic and mesoscopic models. In this study, specific attention is paid to the meshless solution of the UC BVP under plane stress conditions, comparing performances related to the application of linear displacement or periodic boundary conditions (BCs). The effect of the geometry of the UC is also investigated since the BVP is formulated for the two simpliest UCs, according to a displacement-based variational formulation assuming the block indefinitely elastic and the mortar joints as zero-thickness elasto-plastic interfaces. It will be showed that the meshless discretization allows obtaining some advantages with respect to a standard FE mesh. The influence of the UC morphology as well as the BCs on the linear and nonlinear UC macroscopic response is discussed for pure modes of failure. The results can be constructive in view of performing a general Fe·Meshless or Meshless² analysis.     

Large deflections of a cantilever beam subjected to a rotational distributed loading

Large deflection problems of a uniform cantilever beam under a rotational distributed loading are formulated by means of a second order nonlinear integro-differential equation. The problem is numerically solved by considering a uniform rotational distributed load and a linearly varying rotational distributed load along the span of the beam. The details of load deflection curves are presented. Assuming Dirac delta function as a load distribution function along the span of the beam, the present general formulation yields the solution for the problem of a uniform cantilever beam with end rotational concentrated load. The numerical results for this case are found to be in good agreement with existing closed form solutions. As the formulation is general, the problem with nonuniform rotational distributed load of any complexity can be solved following the present numerical procedure which is quite simple, accurate and involves less computational time.     

Banach空间半直线上奇异脉冲积分-微分方程初值问题解的存在性和唯一性

本文在更广泛的情况下,利用Sadovskii不动点定理研究了Banach空间中半直线上一类非线性奇异脉冲积分-微分方程初值问题解的存在性和唯一性,推广并改进了已有文献中的相关结果。     

De-icing by slot injection

Summary We consider the removal of ice from a plate in a cold cross flow by injection of hot fluid through a slot in the plate. De-icing of this sort is required in a number of diverse industrial scenarios, and is particularly relevant to the aviation industry, where the presence of ice on aircraft wings is a major safety hazard. Thin aerofoil theory is used to determine the flow above the injected fluid layer, and this is coupled to flow and energy equations in the injected layer and the ice. The key non-dimensional parameters and ratios in the problem are identified. The result is a nonlinear singular integro-differential equation which is coupled to a convection/diffusion equation and a Stefan condition. Some special cases are discussed and some asymptotic limits are identified. The problem is then solved numerically, and results for a number of different cases are presented.     

Mass transport with enzyme reactions

Summary A mathematical approach is presented for modelling the reactions catalyzed by enzymes attached to the inner surface of a tube through which the substrate solution passes. In a classical paper of Kobayashi and Laidler [4] a theoretical treatment of this problem was considered but the resulting nonlinear equation appeared to be intractable, except for the simple cases. In the present paper, the central integro-differential equation is reduced to an integral Volterra equation. The analysis is extended to different transport situations and to reactions with classical Michaelis-Menten kinetics and with different types of inhibition. Explicit power series representations of the solutions are established for several reaction velocity models. The consistency of the mathematical model used is established by proving the existence and the uniqueness of the continuous solution of the basic integral equation.     

Application of the boundary-domain integral equation in elastic inclusion problems

A boundary-domain integral equation is used to calculate the elastic stress and strain field in a finite or infinite body of isotropic, orthotropic or anisotropic materials characterized with inclusions of arbitrary shapes. Based on the Betti–Rayleigh reciprocal work theorem between the unknown state and a known fundamental solution, the equilibrium of the body with inclusions is formulated in terms of boundary-domain integral equations. The resulting equation involves only the fundamental solution of isotropic medium, and hence the use of complicated fundamental solution for anisotropic materials could be avoided. Numerical examples are given to ascertain the correctness and effectiveness of the boundary-domain integral equation technique for the inclusion problems.