Impact force analysis using the B-spline material point method

In the MPM algorithm, all the particles are formulated in a single-valued velocity field hence the non-slip contact can be satisfied without any contact treatment. However, in some impact and penetration problems, the non-slip contact condition is not appropriate and may even yield unreasonable results, so it is important to overcome this drawback by using a contact algorithm in the MPM. In this paper, the variation of contact force with respect to time caused by the impact is investigated. The MPM using the Lagrange basis function, so causing the cell-crossing phenomenon when a particle moves from one cell to another. The essence of this phenomenon is due to the discontinuity of the gradient of the linear basis function. The accuracy of the results is therefore also affected. The high order B-spline MPM is used in this study to overcome the cell-crossing error. The BSMPM uses higher-order B-spline functions to make sure the derivatives of the shape functions are continuous, so that alleviate the error. The algorithm of MPM and BSMPM has some differences in defining the computational grid. Hence, the original contact algorithm in MPM needs to be modified to be suitable in order to use in the BSMPM. The purpose of this study is to construct a suitable contact algorithm for BSMPM and then use it to investigate the contact force caused by impact. Some numerical examples are presented in this paper, the impact of two circular elastic disks and the impact of a soft circular disk into a stiffer rectangular block. All the results of contact force obtained from this study are compared with finite element results and perform a good agreement, the energy conservation is also considered.

Linearizations for Interpolatory Bases - a Comparison: New Families of Linearizations

One strategy to solve a nonlinear eigenvalue problem $T(\lambda)x=0$ is to solve a polynomial eigenvalue problem (PEP) $P(\lambda)x=0$ that approximates the original problem through interpolation. Then, this PEP is usually solved by linearization. Because of the polynomial approximation techniques, in this context, $P(\lambda)$ is expressed in a non-monomial basis. The bases used with most frequency are the Chebyshev basis, the Newton basis and the Lagrange basis. Although, there exist already a number of linearizations available in the literature for matrix polynomials expressed in these bases, new families of linearizations are introduced because they present the following advantages: 1) they are easy to construct from the matrix coefficients of $P(\lambda)$ when this polynomial is expressed in any of those three bases; 2) their block-structure is given explicitly; 3) it is possible to provide equivalent formulations for all three bases which allows a natural framework for comparison. Also, recovery formulas of eigenvectors (when $P(\lambda)$ is regular) and recovery formulas of minimal bases and minimal indices (when $P(\lambda)$ is singular) are provided. The ultimate goal is to use these families to compare the numerical behavior of the linearizations associated to the same basis (to select the best one) and with the linearizations associated to the other two bases, to provide recommendations on what basis to use in each context. This comparison will appear in a subsequent paper.

On Investigation of Liquid Sloshing in Cylindrical Tanks With Single and Multiply Connected Domains Using Isogeometric Boundary Element Method

This paper explores an isogeometric boundary element method (IGA-BEM) for sloshing problems in cylindrical tanks with single and multiply connected domains. Instead of the Lagrange basis functions used in the standard BEM, the nonuniform rational B-splines (NURBS) basis functions are introduced to approximate the geometries of the problem boundaries and the unknown variables. Compared with the Lagrange basis functions, NURBS basis functions can accurately reconstruct the geometric boundary of analysis domain with almost no error, and all the data information for NURBS basis functions can be directly obtained from the computer-aided design (cad) or computer-aided engineering (cae) commercial software, which implies the modeling process of IGA-BEM is more simple than that of the standard BEM. NURBS makes it possible for the IGA-BEM to realize the seamless connection between cad and cae software with relative higher calculation accuracy than the standard BEM. Based on the weighted residual method as well as the divergence theorem, the IGA-BEM is developed for the single and multiply connected domains, whose boundaries are separately defined in the parameter space by different knot vectors. The natural sloshing frequencies of the liquid sloshing in a circular cylindrical tank with a coaxial or an off-center circular pipe, an elliptical cylindrical tank with an elliptical pipe, a circular cylindrical tank with multiple pipes are estimated with the introduced method by assuming an ideal (inviscid and incompressible) liquid, irrotational small-amplitude sloshing, and the linear free-surface condition. The comparison between the results obtained by the proposed method and those in the existing literatures shows very good agreements, which verifies the proposed model well. Meanwhile, the effects of radius ratio, liquid depth, number, and location of internal pipe (pipes) on the natural sloshing frequency and sloshing mode are analyzed carefully, and some conclusions are outlined finally.

On variable step highly stable 4-stage Hermite-Birkhoff solvers for stiff ODEs

Variable-step (VS) \(4\)-stage \(k\)-step Hermite--Birkhoff (HB) methods of order \(p=(k+1)\), denoted by HB\((p)\), are constructed as a combination of linear \(k\)-step methods of order \((p-2)\) and a two-step diagonally implicit \(4\)-stage Runge--Kutta method of order 3 (TSDIRK3) for solving stiff ordinary differential equations. The main reason for considering this class of formulae is to obtain a set of \(k\)-step methods which are highly stable and are suitable for the integration of stiff differential systems whose Jacobians have some large eigenvalues lying close to the imaginary axis. The approach, described in the present paper, allows us to develop \(L\)-stable \(k\)-step methods of order up to 7 and \(L(\alpha)\)-stable methods of order up to 10 with \(\alpha > 64^\circ\). Fast algorithms are developed for solving confluent Vandermonde-type systems of the new methods in O\((p^2)\) operations to obtain HB interpolation polynomials in terms of generalized Lagrange basis functions. The step sizes of these methods are controlled by a local error estimator. Selected HB(\(p\)) of order \(p\), \(p=4,5,\ldots,9\), compare favorably with existing Cash modified extended backward differentiation formulae, MEBDF(\(p\)), \(p=4,5,\ldots,8\) in solving problems often used to test highly stable stiff ODE solvers on the basis of CPU time, number of steps and error at the endpoint of the integration interval.

Bernstein-Polynomials-Based Highly Accurate Methods for One-Dimensional Interface Problems

A new numerical method based on Bernstein polynomials expansion is proposed for solving one-dimensional elliptic interface problems. Both Galerkin formulation and collocation formulation are constructed to determine the expansion coefficients. In Galerkin formulation, the flux jump condition can be imposed by the weak formulation naturally. In collocation formulation, the results obtained by B-polynomials expansion are compared with that obtained by Lagrange basis expansion. Numerical experiments show that B-polynomials expansion is superior to Lagrange expansion in both condition number and accuracy. Both methods can yield high accuracy even with small value ofN.

Variance-Based Global Sensitivity Analysis via Sparse-Grid Interpolation and Cubature

The stochastic collocation method using sparse grids has become a popular choice for performing stochastic computations in high dimensional (random) parameter space. In addition to providing highly accurate stochastic solutions, the sparse grid collocation results naturally contain sensitivity information with respect to the input random parameters. In this paper, we use the sparse grid interpolation and cubature methods of Smolyak together with combinatorial analysis to give a computationally efficient method for computing the global sensitivity values of Sobol’. This method allows for approximation of all main effect and total effect values from evaluation of f on a single set of sparse grids. We discuss convergence of this method, apply it to several test cases and compare to existing methods. As a result which may be of independent interest, we recover an explicit formula for evaluating a Lagrange basis interpolating polynomial associated with the Chebyshev extrema. This allows one to manipulate the sparse grid collocation results in a highly efficient manner.