Morphing structures: non-linear composite shells with irregular planforms

A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. The treatment of the Dirichlet and Neumann boundary conditions in the new approach is related to the development of high-order boundary conditions with the stencils that include the same or a smaller number of grid points compared to that for the regular 9-point internal stencils. At similar 9-point stencils, the accuracy of the new approach is two orders higher than that for the linear finite elements. The numerical results for irregular domains in Part 2 of the paper also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. Similar to our recent results on regular domains, the order of the accuracy of the new approach for the Poisson equation on irregular domains with square Cartesian meshes is higher than that with rectangular Cartesian meshes. The new approach can be directly applied to other partial differential equations.

Free Vibration Analysis of Curvilinear Quadrilateral Plates by the Differential Quadrature Method

The differential quadrature method (DQM) has been studied for years and it has been shown by many researchers that the DQM is an attractive numerical method with high efficiency and accuracy. The conventional DQM is mostly effective for one‐dimensional and multidimensional problems with geometrically regular domains. But to deal with problems on a triangular domain, we will meet difficulties. In this article we will study how to solve problems on a triangular domain by using DQM combined with the domain decomposition method (DDM). © 2004 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2005

Used to describe the propagation of the electrical potential in heterogeneous cardiac tissue, the FitzHugh-Nagumo monodomain model has attracted numerous attentions to be analyzed. Due to the irregular profile of the cardiac tissue, there has been limitations for the numerical solution on the considered regular domains or approximate irregular domains, where the analytic solution could not be obtained. In this paper, a control volume finite element method is considered to solve the two-dimensional fractional-in-time-space FitzHugh-Nagumo monodomain model with homogeneous Neumann boundary conditions on arbitrary irregular domains with unstructured mesh. By using the linear interpolation shape function methods and matrix transfer techniques, the matrices representation of the fractional Laplacian (-∇ <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ) <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">α/2</sup> are simply obtained. With a mixed explicit-implicit scheme in time, computing the discretised equations with the flux treated implicitly and the source term treated explicitly is transformed to compute matrix-function-vector products. The highlights of this method are that the considered solution domains can be arbitrary, the proposed strategy can be generalised to lower or higher dimensions and the boundary conditions can be generalised to other types. Finally, the efficiency of our method is show-cased by solving fractional-in-space FitzHugh-Nagumo models in two dimensions.

Application of localized meshless methods to 2D shallow water equation problems

In present work, a kind of spectral meshless radial point interpolation (SMRPI) technique is applied to the time fractional nonlinear Schrödinger equation in regular and irregular domains. The applied approach is based on erudite combination of meshless methods and spectral collocation techniques. The point interpolation method with the help of radial basis functions is used to construct shape functions which play as basis functions in the frame of SMRPI. It is proved the scheme is unconditionally stable with respect to the time variable in and also convergent by the order of convergence , . In the current work, the thin plate spline are used as the basis functions and to eliminate the nonlinearity, a simple predictor‐corrector (P‐C) scheme is performed. It is shown that the SMRPI solution, as a complex function, is suitable one for the time fractional nonlinear Schrödinger equation. The results of numerical experiments are compared to analytical solutions to confirm the reliable treatment of these stable solutions. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1043–1069, 2017

High order numerical method for the simulation of Rayleigh–Stokes problem for a heated generalized second grade fluid with fractional derivative on regular and irregular domains

SummaryThis work focuses on the construction of p‐version finite elements that have curve boundaries for C1 problems. Both triangular and quadrilateral elements are constructed based on the C1‐version blending function interpolation methods that are developed in this work and in the literature. Orthogonal hierarchical bases are constructed and subsequently transformed into interpolative nodal bases to facilitate the imposition of boundary conditions and the implementation of C1 conformity on curvilinear domains. Nodal collocation strategies are also studied for improving the numerical performance, and novel nonuniformly distributed nodes, namely, Gauss‐Jacobi (GJ) points, are proposed. For parallelograms and straight‐sided triangular elements, C1 continuity is exactly satisfied between neighboring elements. The difficulty of C1 conformity for elements that have curved boundaries is circumvented by interpolating the normal derivatives at Gauss‐Lobatto nodes. Moreover, with the help of the blending function interpolation method, the bases on edges and the internal modes can differ in terms of approximation order. Therefore, local p‐refinements can be easily performed by these elements. Numerical results demonstrated that these elements are computationally inexpensive and converge fast for problems with regular and irregular domains.

Numerical solution of an inverse source problem for a time-fractional PDE via direct meshless local Petrov–Galerkin method

Efficient polynomial kernel-based method for multi-term time-fractional diffusion systems in regular and irregular domains

The local meshless method based on Pascal polynomial basis functions for solving fourth-order PDEs

Buckling analysis of thin rectangular plates under uniaxial or biaxial compressive point loads by the differential quadrature method

This research presents an intermediate compiler representation that is designed for optimization, and emphasizes the temporary storage requirements and execution schedule of a given computation to guide optimization decisions. The representation is expressed as a dataflow graph that describes computational statements and data mappings within the polyhedral compilation model. The targeted applications include both the regular and irregular scientific domains. The intermediate representation can be integrated into existing compiler infrastructures. A specification language implemented as a domain specific language in C++ describes the graph components and the transformations that can be applied. The visual representation allows users to reason about optimizations. Graph variants can be translated into source code or other representation. The language, intermediate representation, and associated transformations have been applied to improve the performance of differential equation solvers, or sparse matrix operations, tensor decomposition, and structured multigrid methods.

The effect of progressive water surface waves on vertical plane buoyant jets is studied numerically. A three-dimensional numerical model in σ-coordinate is developed to study these problems. Turbulence effects are modeled by a subgrid-scale (SGS) model using the concept of large eddy simulation (LES). The σ-coordinate transformation is introduced to map the irregular physical domain with the wavy free surface and uneven bottom to the regular computational domain with the shape of rectangular prism. The operator splitting method, which splits the solution procedure into the advection, diffusion, and propagation steps, is used to solve the modified Navier-Stokes equations and energy equation. In the study, the heated plane jets are used to validate the numerical model. Distribution of unsteady velocity along centerline and temperature are compaired with the available experimental data. Gaussian and self-similar of buoyant jet mean velocity profile in the Zone of Established Flow (ZEF ) are also observed.

The primary objective of this study is to demonstrate the effectiveness of the radial basis function partition of unity collocation method (RBF-PUM) for solving the fractional Rayleigh-Stokes problem in two dimensions. The time-fractional derivative is formulated in the Riemann–Liouville sense. Initially, the time-fractional derivatives in the governing equation are discretized, followed by the application of RBF-PUM to approximate the spatial derivatives. This method offers notable advantages, including local adaptivity and the ability to adjust node density within each partition. These features help mitigate the ill-conditioning commonly encountered in traditional RBF-based methods. The unconditional stability and convergence of the time-discrete scheme are established using the energy method. To validate the theoretical findings and illustrate the efficiency of RBF-PUM, two numerical experiments are conducted on regular and irregular 2D domains, alongside comparative numerical tests.