A modified numerical manifold method for simulation of finite deformation problem

With many people contributing to its modifications and advancements, the numerical manifold method (NMM) is now recognized as an efficient tool to solve the continuum–discontinuum coupling problem in geotechnical engineering. However, false solutions have been found when modeling finite deformation problems using the original NMM. Based on the finite deformation theory, a modified version of NMM is derived from the weak form of conservation of momentum and the corresponding traction boundary condition. By taking the dual cover system as the displacement approximation, the governing equations of the modified NMM are formulated. A comparison of the governing equations of the original NMM and modified NMM illustrates the reason that the original NMM is not suitable for simulation of finite deformation problems. Three numerical examples are investigated to verify the capability of proposed method to predict static and dynamic finite deformation response. Numerical results show that the modified NMM eliminates the errors caused by large rotation and large strain, and obtains a good agreement with analytical solutions and the finite element method.     

Enriched mixed numerical manifold formulation with continuous nodal gradients for dynamics of fractured poroelasticity

Based on the three-variable Biot model, a numerical manifold model is presented to investigate dynamic responses of the fractured poroelasticity, especially the interaction between hydro-mechanical wave and fracture. The most flexible 3-node triangular mesh serves as the mathematical cover (MC) of the present model regardless of the problem types, shunning difficulties in qualified mesh generation. Continuous nodal gradients and Kronecker-delta property are achieved for the global approximations by constructing the local approximations with a constrained and orthonormalized least square (CO-LS) scheme, presenting more precise effective stress fields and more convenience in implementing boundary conditions for both solid and fluid phases. Incorporating a stick-slip frictional contact model via the augmented Lagrange multiplier method, the present model is capable of accurately predicting the contact phenomenon in fractured poroelasticity. In terms of the energy balance condition, precision and stability of the proposed model in time integration are verified. Fractured porous media involved multiple cracks can be addressed more naturally and conveniently with the present model relative to extended finite element method (XFEM) and phantom node method (PNM). By solving a set of typical porous media problems, the superiority, accuracy and robustness of the present model are verified.     

A high-order numerical method for solving the 2D fourth-order reaction-diffusion equation

Numerical Algorithms - In the present work, orthogonal spline collocation (OSC) method with convergence order O(τ3?α + hr+?1) is proposed for the two-dimensional (2D)...     

Hydraulic fracturing modeling using the enriched numerical manifold method

Recent attempts to solve rock mechanics problems using the numerical manifold method (NMM) have been regarded as fruitful. In this paper, a coupled hydro-mechanical (HM) model is incorporated into the enriched NMM to simulate fluid driven fracturing in rocks. In this HM model, a “cubic law” is employed to model fluid flow through fractures. Several benchmark problems are investigated to verify the coupled HM model. The simulation results agree well with the analytical and experimental results, indicating that the coupled HM model is able to simulate the hydraulic fracturing process reliably and correctly.     

A method of high-order accuracy for the numerical integration of boundary value problems

A finite-difference method for the integration of the linear two-point boundary value problem $$y'' = f(x)y + g(x), y(a) = y_a , y(b) = y_b $$ is constructed. It uses values off′,f″,g′,g″ at the grid points to obtainO(h ⁸) global error, and allows strict global error bounds, while needing only the solution of a tridiagonal system of equations. Numerical examples are presented to demonstrate the practical usefulness of our method.     

A numerical study of singular stress field of 3D cracks

We investigate the stress distribution and the variation of the mode I stress intensity factor along a straight three-dimensional (3D) crack by the finite element method. The results are checked against plane strain theory near the mid-crack and against the 3D theory of Zhu at the free surface. Although Zhu's formulation is not perfect and has some typographical errors. The surface stress distribution of his results are in line with the present study by the finite element method. The stress intensity factors at the free surface are found to be much lower than that at the mid-crack.     

A numerical method for a class of continuous concave programming problems

The problem under consideration consists in maximizing a separable concave objective functional on a class of non-negative Lebesgue integrable functions satisfying a system of linear constraints. The problem is approximated by two sequences of concave separable programming problems with linear constraints. The convergence of the sequences of optimum values of these problems is investigated in the general case and the convergence of the sequences of optimum solutions in a special case. A numerical example is given.     

A thermodynamically consistent numerical method for a phase field model of solidification

A discretization is presented for the initial boundary value problem of solidification as described in the phase-field model developed by Penrose and Fife (1990) [1] and Wang et al. (1993) [2]. These are models that are completely derived from the laws of thermodynamics, and the algorithms that we propose are formulated to strictly preserve them. Hence, the discrete solutions obtained can be understood as discrete dynamical systems satisfying discrete versions of the first and second laws of thermodynamics. The proposed methods are based on a finite element discretization in space and a midpoint-type finite-difference discretization in time. By using so-called discrete gradient operators, the conservation/entropic character of the continuum model is inherited in the numerical solution, as well as its Lyapunov stability in pure solid/liquid equilibria.     

A numerical method for analyzing the quasistationary field above a stratified medium with an axisymmetric elevation

We consider the quasistationary electromagnetic field above a plane-parallel stratified medium with an axisymmetric elevation. The three-dimensional vector problem is reduced to scalar problems in the azimuthal halfplane for the first two field harmonics. A system of one-dimensional integral equations of the first kind is obtained.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 176–181, 1985.     

A stable manifold MCMC method for high dimensions

We combine two important recent advancements of MCMC algorithms: first, methods utilizing the intrinsic manifold structure of the parameter space; then, algorithms effective for targets in infinite-dimensions with the critical property that their mixing time is robust to mesh refinement.     

An implicit one-step method of high-order accuracy for the numerical integration of ordinary differential equations

For a differential equationdx/dt=f(t, x) withf _t (t, x),f _x (t, x) computable, the author presents a new one-step method of high-order accuracy. A rule of controlling the mesh size is given and the method is compared with the Runge-Kutta method in two numerical examples.Dedicated to Professor Dr. Dr. h. c. L. Collatz for his 60^th birthday     

The stressed state caused by a residual strain field in plates with stress concentrators

We obtain an analytic solution of the problem of determining the stressed state caused by a given residual strain field (while taking account of three-dimensional effects) in a round plate with a concentric foreign inclusion. We study the influence of the geometric parameters of the given system, the nonlinearity of the distribution of the given residual strains over the thickness of the plate, and a possible jump in distortion at the surface of contact on the stressed state of the system. We discover an internal boundary-layer effect that is significant in the case when the given residual strain field is strongly gradient.Translated fromMatematicheskie Melody i Fiziko-Mekhanicheskie Polya, Issue 31, 1990, pp. 60–66.     

A high-order non-conforming discontinuous Galerkin method for time-domain electromagnetics

In this paper, we discuss the formulation, stability and validation of a high-order non-dissipative discontinuous Galerkin (DG) method for solving Maxwell’s equations on non-conforming simplex meshes. The proposed method combines a centered approximation for the numerical fluxes at inter element boundaries, with either a second-order or a fourth-order leap-frog time integration scheme. Moreover, the interpolation degree is defined at the element level and the mesh is refined locally in a non-conforming way resulting in arbitrary-level hanging nodes. The method is proved to be stable and conserves a discrete counterpart of the electromagnetic energy for metallic cavities. Numerical experiments with high-order elements show the potential of the method.     

A high-order enrichment strategy for the finite cell method

Thanks to the application of the immersed boundary approach in the finite cell method, the mesh can be defined independently from the geometry. Although this leads to a significant simplification of the mesh generation, it might cause difficulties in the solution. One of the possible difficulties will occur if the exact solution of the underlying problem exhibits a kink inside an element, for instance at material interfaces. In such a case, the solution turns out less smooth – and the convergence rate is deteriorated if no further measures are taken into account. In this paper, we explore a remedy by considering the partition of unity method. The proposed approach allows to define enrichment functions with the help of a high-order implicit representation of the material interface. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling 2D transient heat conduction problems by the numerical manifold method on Wachspress polygonal elements

Due to the use of dual cover systems, i.e., the mathematical cover and the physical cover, the numerical manifold method (NMM) is able to solve physical problems with boundary-inconsistent meshes. Meanwhile, n-gons (n>4) are very impressive, due to their greater flexibility in discretization, less sensitivity to volumetric and shear locking, and better suitability for complex microstructures simulation. In this paper, the NMM, combined with Wachspress-type hexagonal elements, is developed to solve 2D transient heat conduction problems. Based on the governing equations, the NMM temperature approximation and the modified variational principle, the NMM discrete formulations are deduced. The solution strategy to time-dependent global equations and the spatial integration scheme are presented. The advantages of the proposed approach in both discretization and accuracy are demonstrated through several typical examples with increasing complexity. The extension of polygonal elements in unsteady thermal analysis within the NMM is realized.