Convergence for the optimal control problems using collocation at Legendre-Gauss points

The convergence of the novel Legendre-Gauss method is established for solving a continuous optimal control problem using collocation at Legendre-Gauss points. The method allows for changes in the number of Legendre-Gauss points to meet the error tolerance. The continuous optimal control problem is first discretized into a nonlinear programming problem at Gauss collocations by the Legendre-Gauss method. Subsequently, we prove the convergence of the Legendre-Gauss algorithm under the assumption that the continuous optimal control problem has a smooth solution. Compared with those of the shooting method, the single step method, and the general pseudospectral method, the numerical example shows that the Legendre-Gauss method has higher computational efficiency and accuracy in solving the optimal control problem.

Numerical solution to the Falkner-Skan equation: a novel numerical approach through the new rational a-polynomials

The new rational a-polynomials are used to solve the Falkner-Skan equation. These polynomials are equipped with an auxiliary parameter. The approximated solution to the Falkner-Skan equation is obtained by the new rational a-polynomials with unknown coefficients. To find the unknown coefficients and the auxiliary parameter contained in the polynomials, the collocation method with Chebyshev-Gauss points is used. The numerical examples show the efficiency of this method.

A new stress-based topology optimization approach for finding flexible structures

This paper presents a new FE-based stress-related topology optimization approach for finding bending governed flexible designs. Thereby, the knowledge about an output displacement or force as well as the detailed mounting position is not necessary for the application. The newly developed objective function makes use of the varying stress distribution in the cross section of flexible structures. Hence, each element of the design space must be evaluated with respect to its stress state. Therefore, the method prefers elements experiencing a bending or shear load over elements which are mainly subjected to membrane stresses. In order to determine the stress state of the elements, we use the principal stresses at the Gauss points. For demonstrating the feasibility of the new topology optimization approach, three academic examples are presented and discussed. As a result, the developed sensitivity-based algorithm is able to find usable flexible design concepts with a nearly discrete 0 − 1 density distribution for these examples.

A Finite Element Method to Predict the Mechanical Behavior of a Pre-Structured Material Manufactured by Fused Filament Fabrication in 3D Printing

In this paper, a numerical method is proposed to simulate the mechanical behavior of a new polymeric pre-structured material manufactured by fused filament fabrication (FFF), where the filaments are oriented along the principal stress directions. The model implements optimized filament orientations, obtained from the G code by assigning materials references in mesh elements. The Gauss points are later configured with the physical behavior while considering a homogeneous solid structure. The objective of this study is to identify the elastoplastic behavior. Therefore, tensile tests were conducted with different filament orientations. The results show that using appropriate material constants is efficient in describing the built anisotropy and incorporating the air gap volume fraction. The suggested method is proved very efficient in implementing multiplex G code orientations. The elastic behavior of the pre-structured material is quasi-isotropic. However, the anisotropy was observed at the yield point and the ultimate stress. Using the Hill criterion coupled with an experimental tabular law of the plastic flow turns out to be suitable for predicting the response of various specimens.

A powerful numerical technique for treating twelfth-order boundary value problems

In this article, a fast algorithm is developed for the numerical solution of twelfth-order boundary value problems (BVPs). The Haar technique is applied to both linear and nonlinear BVPs. In Haar technique, the twelfth-order derivative in BVP is approximated using Haar functions, and the process of integration is used to obtain the expression of lower-order derivatives and approximate solution for the unknown function. Three linear and two nonlinear examples are taken from literature for checking the convergence of the proposed technique. A comparison of the results obtained by the present technique with results obtained by other techniques reveals that the present method is more effective and efficient. The maximum absolute and root mean square errors are compared with the exact solution at different collocation and Gauss points. The convergence rate using different numbers of collocation points is also calculated, which is approximately equal to 2.

Efficient Numerical Scheme for the Solution of Tenth Order Boundary Value Problems by the Haar Wavelet Method

In this paper, an accurate and fast algorithm is developed for the solution of tenth order boundary value problems. The Haar wavelet collocation method is applied to both linear and nonlinear boundary value problems. In this technqiue, the tenth order derivative in boundary value problem is approximated using Haar functions and the process of integration is used to obtain the expression of lower order derivatives and approximate solution for the unknown function. Three linear and two nonlinear examples are taken from literature for checking validation and the convergence of the proposed technique. The maximum absolute and root mean square errors are compared with the exact solution at different collocation and Gauss points. The experimental rate of convergence using different number of collocation points is also calculated, which is nearly equal to 2.

A Non-Matching Nodes Interface Model with Radial Interpolation Function for Simulating 2D Soil–Structure Interface Behaviors

In this paper, the meshless method is extended to simulate the interaction between soil and structure through 2D finite element (FE) model. The background mesh line shared by each surface of interface is introduced for Gauss points’ generation and interpolation. Thus, instead of a series of interface elements, the whole soil–structure interface can be presented by an arbitrary number of nodes with flexible distribution on the contacting surfaces. The radial basis function (RBF) is introduced as the interpolation function to obtain the displacement of each Gauss points by surrounding the nodes along the surfaces. The research of shape parameters and the rate of convergence are also conducted. With non-matching nodes interface, the soil and structure zone can be modeled independently and connected with the proposed non-matching nodes interface flexibly and effectively. Furthermore, the proposed method can easily enable cross-scale modeling, which has high utility for enabling refined analysis without excessively increasing the computational costs. In addition, under the standard finite element framework, nonlinear constitutive models can be employed to capture the complex behaviors at the interface. Several simulations are presented to demonstrate the high flexibility, extensive applicability and precision of the proposed non-matching nodes interface based on meshless method.

Higher order multipoint flux mixed finite element methods on quadrilaterals and hexahedra

We develop higher order multipoint flux mixed finite element (MFMFE) methods for solving elliptic problems on quadrilateral and hexahedral grids that reduce to cell-based pressure systems. The methods are based on a new family of mixed finite elements, which are enhanced Raviart–Thomas spaces with bubbles that are curls of specially chosen polynomials. The velocity degrees of freedom of the new spaces can be associated with the points of tensor-product Gauss–Lobatto quadrature rules, which allows for local velocity elimination and leads to a symmetric and positive definite cell-based system for the pressures. We prove optimal [Formula: see text]th order convergence for the velocity and pressure in their natural norms, as well as [Formula: see text]st order superconvergence for the pressure at the Gauss points. Moreover, local postprocessing gives a pressure that is superconvergent of order [Formula: see text] in the full [Formula: see text]-norm. Numerical results illustrating the validity of our theoretical results are included.

The Equilibrium Cell-Based Smooth Finite Element Method for Shakedown Analysis of Structures

This paper presents a novel equilibrium formulation, that uses the cell-based smoothed method and conic programming, for limit and shakedown analysis of structures. The virtual strains are computed using straining cell-based smoothing technique based on elements of discretized mesh. Fictitious elastic stresses are also determined within the framework of finite element method (CS-FEM)-based Galerkin procedure, and equilibrium equations for residual stresses are satisfied in an average sense at every cell-based smoothing cell. All constrains are imposed at only one point in the smoothing domains, instead of Gauss points as in a standard FEM-based procedure. The resulting optimization problem is then handled using the highly efficient solvers. Various numerical examples are investigated, and obtained solutions are compared with available results in the literature.