International Journal of Computational Methods

A massively parallel direct numerical solution procedure for the turbulent Ekman layer is presented. The simulations study the dynamics of turbulence in this flow by solving the incompressible Navier–Stokes equations with Coriolis and buoyancy terms. The governing equations are integrated via a semi-implicit time advancement algorithm which is massively parallelized using the Portable, Extensible Toolkit for Scientific Computation (PETSc) libraries. Accuracy of the numerical scheme was validated by comparisons of simulation results with the hydrodynamic linear stability theory for Poiseuille flow. Two cases are presented to demonstrate the capabilities of the code: (a) a neutrally stable case of Reynolds number, Re = 400 and (b) an unstably stratified case at Re = 1,000 requiring very high resolution in all coordinate directions. Results indicate that the scalability is not limited by the overall size of the problem, but rather by the number of mesh points per processor. Strong scaling is demonstrated for both cases with as few as 10,000 unknowns per processor.

We propose a novel method for a solution of a system of linear equations with the non-negativity condition. The method is based on the Tikhonov functional and has better accuracy and stability than other well-known algorithms.

In this paper, the spectral collocation method is investigated for the numerical solution of multi-order Fractional Differential Equations (FDEs). We choose the orthogonal Jacobi polynomials and set of Jacobi Gauss–Lobatto quadrature points as basis functions and grid points respectively. This solution strategy is an application of the matrix-vector-product approach in spectral approximation of FDEs. The fractional derivatives are described in the Caputo type. Numerical solvability and an efficient convergence analysis of the method have also been discussed. Due to the fact that the solutions of fractional differential equations usually have a weak singularity at origin, we use a variable transformation method to change some classes of the original equation into a new equation with a unique smooth solution such that, the spectral collocation method can be applied conveniently. We prove that after this regularization technique, numerical solution of the new equation has exponential rate of convergence. Some standard examples are provided to confirm the reliability of the proposed method.

This paper studies T-stress problem for multiple cracks in an infinite plate with the remote loading. After some manipulations, the problem can be modeled by dislocation distributions along the crack faces. A singular integral equation is formulated for the problem, where the unknowns are the dislocation distribution along the crack faces. The SIFs (stress intensity factor) can be evaluated from the solution of singular integral equation. From a definition for T-stress in the crack back position method and the solution of the singular integral equation, the T-stresses at crack tips can be evaluated accordingly. An explicit formula for the T-stress is provided. Several numerical examples are provided. Accuracy of computation is examined by an example.

In isogeometric analysis, the tensor-product form of Nonuniform Rational B-spline (NURBS) represents spline surfaces. Due to the nature of the tensor-product, the local refinement in isogeometric analysis has many issues to be resolved. Attempts have been made in this regard, such as T-splines and hierarchical approaches. In this work, a local refinement method for isogeometric analysis based on a superimposing concept is proposed. Local refinements are performed by superimposing hierarchically-created finer overlay meshes onto the regions of high error rather than a change of analysis basis (from NURBS to some other spline space). To employ the superimposing concept as a local refinement strategy in isogeometric analysis, a hierarchical framework to construct overlay meshes is developed, and compatibility conditions across the interfacial boundaries of different levels of meshes is discussed. Through numerical examples, the effectiveness and validity of the proposed method are demonstrated.

Many triangular elements have been formulated for bending plate analysis so far. In this type of element, researchers generally considered the same distance between the side nodes. The present study will show that these types of node locations are not the best. To clarify the effect of side node locations, three new triangular elements will be formulated for the analysis of thin bending plates. These are nonconforming elements and have 10, 15, and 21 degrees of freedom, respectively. Merits and disadvantages of these proposed elements will be revealed by solving different structures. Optimal side nodes' location will be positioned by changing the place of element nodes. Numerical tests will confirm that element accuracy in the best possible situation is much better than the state in which side nodes have the same distance. Moreover, this study will reveal that the presented optimal locations of nodes have no significant dependence on the load cases, boundary conditions, distortion of element shape, and variation of the meshes.

In this paper, we investigate spectral norms for circulant-type matrices, including circulant, skew-circulant, and g-circulant matrices. The entries are product of Binomial coefficients with Harmonic numbers. We obtain explicit identities for these spectral norms. Employing these approaches, we list some numerical tests to verify our results.

A refined and simple shear deformation theory for thermal buckling of solar functionally graded plate (SFGP) resting on two-parameter Pasternak's foundations is developed. The displacement field is chosen based on assumptions that the in-plane and transverse displacements consist of bending and shear components, and the shear components of in-plane displacements give rise to the parabolic variation of shear strain through the thickness in such a way that shear stresses vanish on the plate surfaces. Therefore, there is no need to use shear correction factor. The number of independent unknowns of present theory is four, as against five in other shear deformation theories. It is assumed that the material properties of the plate vary through the thickness of the plate as a power function. The neutral surface position for such plate is determined, and the present plate theory based on exact neutral surface position is employed to derive the governing stability equations. The nonlinear strain-displacement relations are also taken into consideration. The boundary conditions for the plate are assumed to be simply supported in all edges. Closed-form solutions are presented to calculate the critical buckling temperature, which are useful for engineers in design. The effects of the foundation parameters, plate dimensions, and power law index are presented comprehensively for the thermal buckling of solar functionally graded plates.

In this paper, we present some three-point Newton-type iterative methods without memory for solving nonlinear equations by using undetermined coefficients method. The order of convergence of the new methods without memory is eight requiring the evaluations of three functions and one first-order derivative in per full iteration. Hence, the new methods are optimal according to Kung and Traubs conjecture. Based on the presented methods without memory, we present two families of Newton-type iterative methods with memory. Further accelerations of convergence speed are obtained by using a self-accelerating parameter. This self-accelerating parameter is calculated by the Hermite interpolating polynomial and is applied to improve the order of convergence of the Newton-type method. The corresponding R-order of convergence is increased from 8 to 9, and 10. The increase of convergence order is attained without any additional calculations so that the two families of the methods with memory possess a very high computational efficiency. Numerical examples are demonstrated to confirm theoretical results.

We adopted operator splitting method to solve the governing equations of explosion and impact problems, and used fuzzy interface treatment to crack 3D multi-material interface difficulties. Then based on message passing interface (MPI), the data dependence and relevance between adjacent subdomains in parallel computing of the Eulerian method were studied and analyzed. At last, we numerically simulated blast in air and shaped charge jet using PMMIC-3D hydrocode, and performed empirical and experimental comparisons. The results show that the cell spatial step has important influence on computational accuracy, and the calculating result with smaller cell spatial step is close to the empirical formula one. Therefore, it is necessary to develop parallel computing to enlarge the calculation scale at smaller cell spatial step for the explosion and impact issues.

A 3D discrete element model (DEM model) named cubic arranged discrete element model is proposed. The model treats the interaction between two connective discrete elements as an equivalent "beam" element. The spring constants between two connective elements are obtained based on the equivalence of strain energy stored in a unit volume of elastic continuum. Following that, the discrete element model proposed and its algorithm are implemented into the in-house developed code. To test the accuracy of the DEM model and its algorithm, the vibration process of the block, a homogeneous plate and laminated plate under impact loading are simulated in elastic range. By comparing the results with that calculated by using LS-DYNA, it is found that they agree with each other very well. The accuracy of the DEM model and its algorithm proposed in this paper is proved.

An edge-based smoothed tetrahedron radial point interpolation method (ES-T-RPIM) is formulated for the 3D acoustic problems, using the simplest tetrahedron mesh which is adaptive for any complicated geometry. In present ES-T-RPIM, the gradient smoothing operation is performed with respect to each edge-based smoothing domain, which is also serving as building blocks in the assembly of the stiffness matrix. The smoothed Galerkin weak form is then used to create the discretized system equations. The acoustic pressure is constructed using radial point interpolation method, and two typical schemes of selecting nodes for interpolation using RPIM have been introduced in detail. It turns out that the ES-T-RPIM provides an ideal amount of softening effect, and significantly reduces the numerical dispersion error in low- to mid-frequency range. Numerical examples demonstrate the superiority of the ES-T-RPIM for 3D acoustic analysis, especially at mid-frequency.

Simulation of dynamic adhesive peeling problems at small scales has attracted little attention so far. These problems are characterized by a highly nonlinear response. Accurate and stable time integration schemes are required for simulation of dynamic peeling problems. In the present work, a composite time integration scheme is proposed for the simulation of dynamic adhesive peeling problems. It is shown through numerical examples that the proposed scheme remains stable and also has some gain in accuracy. The performance of the scheme is compared with two collocation-based schemes, i.e., Newmark scheme and Bathe composite scheme. It is shown that the proposed scheme and Bathe composite scheme perform equally. However, the proposed scheme adds very little to the computational cost of Newmark scheme. Through a numerical simulation of the peeling of a gecko spatula from a rigid substrate it is shown that the proposed scheme and the Bathe composite scheme are able to simulate the complete peeling process for given time step whereas the Newmark scheme diverges. It is also shown that the maximum pull-off force is within the range reported in the literature.