Analysis and Evaluation of Piecewise Linear Systems With Coulomb Friction Using a Hybrid Symbolic-Numeric Computational Method

A general formulation of piecewise linear systems with discontinuous force elements is provided in this paper. It has been demonstrated that this class of nonlinear systems is of great importance due to their ability to accurately model numerous scientific and engineering phenomena. Additionally, it is shown that this class of nonlinear systems can demonstrate a wide spectrum of nonlinear motions and in fact, the phenomenon of weak chaos is observed in a mechanical assembly for the first time. Despite such importance, efficient methods for fast and accurate evaluation of piecewise linear systems’ responses are lacking and the methods of the literature are either incompatible, very slow, very inaccurate, or bear a combination of the aforementioned deficiencies. To overcome this shortcoming, a novel symbolic-numeric method is presented in this paper that is able to obtain the analytical response of piecewise linear systems with discontinuous elements in an efficient manner. Contrary to other efficient methods that are based on stationary steady state dynamics, this method will not experience failure upon the occurrence of complex motion and is able to capture the entirety of the dynamics.

Three-Dimensional Elastodynamic Analysis Employing Partially Discontinuous Boundary Elements

Compared with continuous elements, discontinuous elements advance in processing the discontinuity of physical variables at corner points and discretized models with complex boundaries. However, the computational accuracy of discontinuous elements is sensitive to the positions of element nodes. To reduce the side effect of the node position on the results, this paper proposes employing partially discontinuous elements to compute the time-domain boundary integral equation of 3D elastodynamics. Using the partially discontinuous element, the nodes located at the corner points will be shrunk into the element, whereas the nodes at the non-corner points remain unchanged. As such, a discrete model that is continuous on surfaces and discontinuous between adjacent surfaces can be generated. First, we present a numerical integration scheme of the partially discontinuous element. For the singular integral, an improved element subdivision method is proposed to reduce the side effect of the time step on the integral accuracy. Then, the effectiveness of the proposed method is verified by two numerical examples. Meanwhile, we study the influence of the positions of the nodes on the stability and accuracy of the computation results by cases. Finally, the recommended value range of the inward shrink ratio of the element nodes is provided.

Boundary element analysis for elasticity problems using expanding element interpolation method

Purpose This paper aims to propose a boundary element analysis of two-dimensional linear elasticity problems by a new expanding element interpolation method. Design/methodology/approach The expanding element is made up based on a traditional discontinuous element by adding virtual nodes along the perimeter of the element. The internal nodes of the original discontinuous element are referred to as source nodes and its shape function as raw shape function. The shape functions of the expanding element constructed on both source nodes and virtual nodes are referred as fine shape functions. Boundary variables are interpolated by the fine shape functions, while the boundary integral equations are collocated on source nodes. Findings The expanding element inherits the advantages of both the continuous and discontinuous elements while overcomes their disadvantages. The polynomial order of fine shape functions of the expanding elements increases by two compared with their corresponding raw shape functions, while the expanding elements still keep independence to each other as the original discontinuous elements. This feature makes the expanding elements able to naturally and accurately interpolate both continuous and discontinuous fields. Originality/value Numerical examples are presented to verify the proposed method. Results have demonstrated that the accuracy, efficiency and convergence rate of the expanding element method.

Numerical Studies of Nonlinear Gearing Models Using Bond Graph Method

The present paper is dedicated to computer simulations performed using a numerical model of a one-stage gear. The motion equations were derived utilizing the bond graph method. The formulated model takes into consideration the variable stiffness of toothings as well as an inter-tooth clearance which has been represented via discontinuous elements with so called dead zones. As a result of these assumptions, the nonlinear model was obtained which enables representation of the dynamic phenomena of the considered gear. In the paper, an influence of errors of gear wheels’ co-operation on the character of excited dynamic phenomena was studied. The methodology of the analyses consists in utilization of the following tools: color maps of distribution of the maximal Lapunov coefficient and bifurcation diagrams. Based upon them, the parameters were determined, for which the Poincare portrait represents a structure of the chaotic attractor. For the identified attractors, the initial attractors were calculated numerically - which along with the changes of the control parameters are subjected to multiplication, stretching or rotation.

Assessment of a Method to Model 2D Discontinuities with Enriched Finite Elements and Level Sets

The computational treatment of discontinuities within the framework of the finite element method (FEM) is a requirement for simulations of fracturing of solids and has become a challenging topic in computational mechanics. Particularly popular amongst the advanced schemes is the extended finite element method (XFEM) which is based on enrichment of shape functions and falls within the framework of the partition of unity method. Because there is no simple way to track the interface of discontinuities, the computer implementation of the XFEM is not as straightforward as the FEM. One method to solve the interface tracking problem is the level set method (LSM) which introduces another partial differential equation. The level set equation describes the change of an interface due to a known velocity field. To obtain its solution, the FEM can also be employed. This contribution investigates the XFEM-LSM technique with element enrichment and the integration of discontinuous elements for the modelling of cracks or material interfaces. Numerical experiments illustrate the capabilities and accuracy of the resulting formulation.

A global finite-element shallow-water model supporting continuous and discontinuous elements

This paper presents a novel nodal finite-element method for either continuous and discontinuous elements, as applied to the 2-D shallow-water equations on the cubed sphere. The cornerstone of this method is the construction of a robust derivative operator that can be applied to compute discrete derivatives even over a discontinuous function space. A key advantage of the robust derivative is that it can be applied to partial differential equations in either a conservative or a non-conservative form. However, it is also shown that discontinuous penalization is required to recover the correct order of accuracy for discontinuous elements. Two versions with discontinuous elements are examined, using either the g1 and g2 flux correction function for distribution of boundary fluxes and penalty across nodal points. Scalar and vector hyperviscosity (HV) operators valid for both continuous and discontinuous elements are also derived for stabilization and removal of grid-scale noise. This method is validated using four standard shallow-water test cases, including geostrophically balanced flow, a mountain-induced Rossby wave train, the Rossby–Haurwitz wave and a barotropic instability. The results show that although the discontinuous basis requires a smaller time step size than that required for continuous elements, the method exhibits better stability and accuracy properties in the absence of hyperviscosity.