International Journal of Computational Methods

This paper proposes a combined discrete–finite element model to investigate the dynamic behavior of ballasted railway tracks. The discrete element method (DEM) is adopted to model the discrete ballast materials. The shapes of ballast particles resembled clumps of overlapping spheres which are obtained by the growth of spheres inside convex polyhedrons. The finite element method (FEM) is used to analyze the continuous embankment and foundation. The transmission between DEM and FEM at the ballast-embankment interface is processed according to the interaction force based on the principle of virtual work. The dynamic behavior of ballasted railway track under cyclic loading is simulated with the developed DEM–FEM model. The settlement of the sleeper and the deformation of the embankment and foundation, the force chains in the ballast and stress distributions in the embankment and foundation are obtained. The developed model is helpful in better understanding the mechanical characteristics of ballasted railway tracks.

A Goal-Oriented and Model-Constrained Optimization (GOMCO) approach was proposed as a discriminant technique for the Variational Geomano Method (VGM) in the coefficient determinations of variational multiscale Unresolved-Scale (URS) model in steady Stokes equations. Numerical implementations using both linear and nonlinear models were performed with both the GOMCO and VGM. Numerical results show that the coefficients determined by the GOMCO are scale-invariant, while they are scale-variant by the VGM. The GOMCO technique is found to be more appropriate for coefficient determinations in steady Stokes equations, as the VGM is sensitive to the computing procedures. Moreover, the GOMCO could provide reliable coefficients for the URS model.

A POD-based variational multiscale method (VMM) for large eddy simulation (LES) of turbulent channel flows is proposed. The incompressible Navier–Stokes equations are solved using a residual-based VMM technique, and the residual-based model coefficient is adopted from Taylor et al. [1998] “Finite element modeling of blood flow in arteries,” Comput. Methods in Appl. Mech. Eng. 158(1–2), 155–196. A generalized-$\alpha$ method is applied for the temporal integration. Numerical results show that four or higher wall-normal modes need to be applied for reproducing most of the turbulent characteristics. Results of large vortex structures and the low velocity streaks prove that the turbulent channel flows are well estimated by the proposed approach. The statistical analysis of the turbulent velocities confirms its reliability in turbulent channel flows.

Compared with the traditional Smoothed Particle Hydrodynamics (SPH), Finite Particle Method (FPM) has higher accuracy for boundary region. However, there are still two inherent defects which are the time consuming and the numerical instability in FPM. In this paper, a high-order algorithm based on the Taylor series expansion and the matrix theory is proposed and the corresponding particles selected mode is discussed. It is validated that the algorithm has higher-order accuracy than the previous low-order improvement algorithm for FPM. Further, transient heat conduction examples have been discussed to verify the feasibility and effectiveness of the new algorithm.

An efficient parallel elastoplastic reanalysis method is suggested. The main backbone of the suggested method is based on combined approximation (CA) reanalysis. GPU parallel computation is used to accelerate assembling the stiffness matrix. Assembling process is divided into the offline part for strain matrix and online part for element stiffness matrix, which makes the structure of the program more reasonable and efficient. Pseudo elastic analysis is introduced and extended to load increment method to make the CA method more feasible. The numerical examples show that the suggested method can improve the efficiency of elastoplastic analysis significantly and the accuracy of results can also be ensured.

Based on the edge-based smoothed FEM (ES-FEM) and the partition of unity, the first major items of Williams’ series for the displacement field near the crack tip are incorporated in the test and trial function space, resulting in the enriched ES-FEM formulation, eEF-FEM. The eES-FEM does not differentiate any shape functions, avoiding the treatment of the $1/r$ singularity in computing the stiffness matrix. The complexity of computation is accordingly reduced. Meanwhile, it is pointed out that the variational foundation of the eES-FEM is the generalized Galerkin method. Typical numerical examples are analyzed, suggesting that the results of the eES-FEM are much better than either FEM or ES-FEM.

In this paper, we present a finite element scheme with Crank–Nicolson method for solving nonlocal parabolic problems involving the Dirichlet energy. We discuss the well-posedness of the weak formulation at continuous as well as at discrete levels. We derive a priori error estimates for both semi-discrete and fully-discrete formulations. Results based on usual finite element method are provided to confirm the theoretical estimates.

This paper presents a simple yet efficient method for the topology optimization of continuum structures considering interval uncertainties in loading directions. Interval mathematics is employed to equivalently transform the uncertain topology optimization problem into a deterministic one with multiple load cases. An efficient soft-kill bi-directional evolutionary structural optimization (BESO) method is proposed to solve the problem, which only requires two finite element analyses per iteration for each external load with directional uncertainty regardless of the number of the multiple load cases. The presented algorithm leads to significant computational savings when compared with Monte Carlo-based optimization (MCBO) algorithms. A series of numerical examples including symmetric and nonsymmetric loading variations demonstrate the considerable improvement of computational efficiency of the proposed approach as well as the significance of including uncertainties in topology optimization when to design a structure. Optimums obtained from the proposed algorithm are verified by MCBO method.

Three-dimensional static response of all-around simply supported functionally graded square/rectangular laminates is presented here based on higher order shear-normal (HOSNT) deformation theory and semi-analytical approach. Modulus of elasticity is assumed to be varied according to power law through the thickness of laminate and other material properties are assumed to be constant over the domain. Semi-analytical approach consists of defining two-point boundary value problem (BVP) in the thickness direction. It involves displacements and transverse stresses as primary degrees of freedoms (DOFs) and therefore, general stress boundary conditions can be applied on both top and bottom surface of laminate during the numerical solution. Whereas, in HOSNT, only displacements are considered as primary DOFs and Taylor’s series are used to expand primary DOFs through thickness direction which helps to take into account of transverse cross-sectional deformation modes. Minimum potential energy is used in HOSNT to derive equilibrium and boundary equation and Navier’s solutions techniques is used to obtain closed form solution. The efficiency of the developed formulation in terms of accuracy and simplicity is examined by various numerical examples and compared with available results based on shear deformation theories.

We present a fully second order IMplicit/EXplicit (IMEX) time integration technique for solving incompressible multi-phase flow problems. A typical incompressible multi-phase flow model consists of the Navier–Stokes equations plus an interface dynamics equation (e.g., the level set equation). Our IMEX strategy is applied to such a model in the following manner. The hyperbolic terms of the Navier–Stokes equations together with the interface dynamics equation are solved explicitly (Explicit Block) making use of the well-understood explicit numerical schemes [Leveque, R. J. [1998] Finite Volume Methods for Hyperbolic Problems, “Texts in Applied Mathematics”, (Cambridge University Press); Thomas, J. W. [1999] Numerical Partial Differential Equations II (Conservation Laws and Elliptic Equations), “Texts in Applied Mathematics” (Springer-Verlag, New York)]. On the other hand, the nonhyperbolic (stiff) parts of the flow equations are solved implicitly (Implicit Block) within the framework of the Jacobian-Free Newton Krylov (JFNK) method [Knoll, D. A. and Keyes, D. E. [2004] Jacobian-free Newton Krylov methods: A survey of approaches and applications. J. Comput. Phys. 193, 357–397; Saad, Y. [2003] Iterative Methods for Sparse Linear Systems (Siam); Kelley, C. T. [2003] Solving Nonlinear Equations with Newton’s Method (Siam)]. In our algorithm implementation, the explicit block is embedded in the implicit block in a way that it is always part of the nonlinear function evaluation. In this way, there exists a continuous interaction between the implicit and explicit algorithm blocks meaning that the improved solutions (in terms of time accuracy) at each nonlinear iteration are immediately felt by the explicit block and the improved explicit solutions are readily available to form the next set of nonlinear residuals. This continuous interaction between the two algorithm blocks results in an implicitly balanced algorithm in that all nonlinearities due to coupling of different time terms are converged with the desired numerical time accuracy. In other words, we obtain a self-consistent IMEX method that eliminates the possible order reductions in time convergence that is quite common in certain types of nonlinearly coupled systems. We remark that an incompressible multi-phase flow model can be a highly nonlinearly coupled system with the involvement of very stiff surface tension source terms. These kinds of flow problems are difficult to tackle numerically. In other words, highly nonlinear surface terms may remain unconverged leading to time inaccuracies or time order reductions to the first order even though the overall numerical scheme is designed as high order (second-order or higher) [Sussman, M. and Ohta, M. [2009] A stable and efficient method for treating surface tension in incompressible two-phase flow, SIAM J. Sci. Comput. 31(4), 2447–2471; Zheng, W., Zhu, B., Kim, B. and Fedkiw, R. [2015] A new incompressibility discretization for a hybrid particle MAC grid representation with surface tension, J. Comput. Phys. 280, 96–142]. These and few more issues are addressed in this paper. We have numerically tested our newly proposed scheme by solving several multi-phase flow settings such as an air bubble rising in water, a Rayleigh–Taylor instability problem that is initiated by placing a heavy fluid on top of a lighter one, and a droplet problem in which a water droplet hits the pool of water. Our numerical results show that we have achieved the second-order time accuracy without any order reductions. Moreover, the interfaces between the fluids are captured reasonably well.

Computational homogenization provides an excellent tool for the design of composite materials. In the current work, a computational approach is presented that is capable of estimating the elastic and rate-independent plastic constitutive behavior of metal matrix composites using finite element models of representative volume elements (RVEs) of the composite material. For this purpose, methodologies for the generation of three-dimensional computational microstructures, size determination of RVEs and the homogenization techniques are presented. Validation of the approach is carried out using aluminum–alumina composite samples prepared using sintering technique. Using the homogenized material response, effective constitutive models of the composite materials have been determined.

In this work, a numerical scheme based on artificial compressibility formulation of a phase-field model is developed for simulating two-phase incompressible flow problems. The coupled nonlinear systems composed of the incompressible Navier–Stokes equations and volume preserving Allen–Cahn-type phase-field equation are recast into conservative form with source terms, which are suited to implement high-resolution schemes originally developed for hyperbolic conservation laws. The Boussinesq approximation is used to account for the buoyancy effect in flow with small density difference. The fifth-order weighted essentially nonoscillatory (WENO) scheme is used for discretizing the convective terms while dual-time stepping (DTS) technique is used for obtaining time accuracy at each physical time step. Beam–Warming approximate factorization scheme is utilized to obtain block tridiagonal system of equations in each spatial direction. The alternating direction implicit (ADI) algorithm is used to solve the resulting system of equations. The performance of the method is demonstrated by its application to some 2D and 3D benchmark viscous two-phase flow problems.