S.C. Fan

A comprehensive unified set of single-step algorithms with controllable dissipation for dynamics Part I. Formulation

A new class of time integration algorithms for dynamics is presented. The algorithms are based on finite element in time. The equations of motion are expressed in first-order form and the variables are interpolated via time polynomials. The unknown nodal variables are solved by the generalized Galerkin method. Algorithms of any desired order of time accuracy can be achieved by proper choice of the finite element interpolation in time. The salient feature of the algorithms is that it can be either asymptotic annihilating or non-dissipative. Besides, any desired algorithmic damping between these two extreme cases could be obtained. The algorithms are analysed in detail. The spectral characteristics equivalent to the upper diagonal, diagonal of the Pade approximations and other rational approximations are given.

A comprehensive unified set of single-step algorithms with controllable dissipation for dynamics Part II. Algorithms and analysis

In Part I, a comprehensive unified set of single-step algorithms with controllable dissipation based on the generalized Galerkin formulation was presented. In this paper, the details of four lower-order algorithms using parametrized temporal finite element are presented. The analyses of the numerical stability, dissipation and dispersion are also provided. In comparison with the established algorithms, the present algorithms facilitate controllable dissipation and yield results of higher-order accuracy. Examples are also included to illustrate the validity of the algorithms.

Meshless formulation using NURBS basis functions for eigenfrequency changes of beam having multiple open-cracks

Abstract This paper presents a meshless formulation using non-uniform rational B-spline (NURBS) basis functions, and its applications to evaluate natural frequencies of a beam having multiple open-cracks. Node-based NURBS basis functions are used to construct the approximation function. The characteristic differentiability of the NURBS basis functions allows it to represent a function having specific degrees of smoothness and/or discontinuity. The discontinuity can be incorporated simply by assigning multiple knots at those locations. Hence, it can yield exact solutions having interior discontinuous derivatives. These advantages of NURBS are well known, and have been used extensively in graphical approximation of geometrical surfaces. However, it is seldom used in other engineering applications. To model the multiple open-cracks in a beam, quartic NURBS basis functions are employed and quadruplicate knots are assigned at the crack locations. Hence, it is capable to model the abrupt changes of slope (the first derivative of displacement) across a crack. In the present applications, additional equivalent massless rotational springs are inserted at the crack locations to represent the local flexibility caused by the cracks. As such, the cracked beam can be treated in the usual manner as a continuous beam. By adopting the meshless Petrov–Galerkin formulation, a generalized stiffness matrix for the cracked beam can be derived. Compared to the conventional finite element method, the present method does not require a finite element mesh for the purposes of interpolation and numerical integration. The advantages and effectiveness of the present method is illustrated in solving the eigenfrequencies of a beam having multiple open-cracks of different depths.

Spline shell element and plane-wave approximation for dynamic response of submerged structures

Abstract This paper presents a performance study on the plane-wave approximation for dynamic responses of shell structures using spline finite elements, in particular, the interaction between submerged shell structures and the surrounding infinite fluid. The shell structures are discretized by using quadrilateral spline shell elements. The element is doubly curved. Each element consists of nine primary nodes and eight auxiliary nodes, and has a total of 63 degrees of freedom. The inter-element is C 1 continuous. A non-reflecting boundary based on acoustic approximations is used to model the infinite fluid medium. Two examples are presented. Results demonstrated good performance.

Analysis of complicated plates by a nine-node spline plate element

Abstract The analysis of complicated plates using a newly developed nine-node quadrilateral spline plate element is presented. The element is based on Kirchhoffs plate theory with biquadratic Lagrangian shape functions for the transformation of geometry and B -spline shape functions for the interpolation of displacements. The accuracy and robustness of the element are checked by using a set of test problems that are capable of detecting inaccuracies due to shape distortion, large aspect ratio and other irregularities. Numerical examples on complicated plate structures such as plates with interior openings or mixed boundary conditions are reported to demonstrate the practical usefulness of the element.

Extrapolated Galerkin time finite elements

A new time finite-element method based on the extrapolation technique and the Galerkin time finite-element method is presented. In this method, the second-order governing differential equations of motion for dynamic problems are rewritten as a set of first order differential equations in state space. The standard Galerkin method is then employed for the temporal discretization. The algorithm is first-order accurate only. Based on the first-order Galerkin time finite-element formulation, the extrapolation technique is introduced to improve the order of accuracy. It is achieved by expressing the numerical amplification matrix of higher-order algorithm as a linear combination of the basic amplification matrices evaluated at selected instances of time. The matrices are combined with different weighting factors. The pairs of the selected instance of time and the corresponding weighting factors are free parameters. Unconditionally stable higher-order accurate formulations can be derived by properly choosing the free parameters. Algorithms up to fourth-order accurate are presented in this paper. Detailed analyses on stability, numerical dissipation and numerical dispersion are also given. Comparisons of the present algorithms with some well-known time-integration methods are presented to demonstrate the versatility of the present method, in particular its accuracy in the higher-order formulations.

Modeling of Combined Impact and Blast Loading on Reinforced Concrete Slabs

EXPLOSIVE DEVICES REPRESENT A SIGNIFICANT THREAT TO MILITARY AND CIVILIAN STRUCTURES. SPECIFIC DESIGN PROCEDURES HAVE TO BE FOLLOWED TO ACCOUNT FOR THIS AND ENSURE BUILDINGS WILL HAVE THE CAPACITY TO RESIST THE IMPOSED PRESSURES. SHRAPNEL CAN ALSO BE PRODUCED DURING EXPLOSIONS AND THE RESULTING IMPACTS CAN WEAKEN THE STRUCTURE, REDUCING ITS CAPACITY TO RESIST THE BLAST PRESSURE WAVE AND POTENTIALLY CAUSING FAILURES TO OCCUR. EXPERIMENTS WERE PERFORMED BY THE DEFENCE SCIENCE AND TECHNOLOGY AGENCY (DSTA) OF SINGAPORE TO STUDY THIS COMBINED LOADING PHENOMENON. SLABS WERE PLACED ON THE GROUND AND LOADED WITH APPROXIMATELY 9 KG TNT CHARGES AT A STANDOFF OF 2.1 M. SPHERICAL STEEL BALL BEARINGS WERE USED TO REPRODUCE THE SHRAPNEL LOADING. LOADING AND DAMAGE CHARACTERISTICS WERE RECORDED FROM THE EXPERIMENTS. A FINITE ELEMENT ANALYSIS (FEA) MODEL WAS THEN CREATED WHICH COULD SIMULATE THE EFFECT OF COMBINED SHRAPNEL IMPACTS AND BLAST PRESSURE WAVES IN REINFORCED CONCRETE SLABS, SO THAT ITS RESULTS COULD BE COMPARED TO EXPERIMENTAL DATA FROM THE BLAST TESTS. QUARTER MODELS OF THE EXPERIMENTAL CONCRETE SLABS WERE BUILT USING LS-DYNA. MATERIAL MODELS AVAILABLE IN THE SOFTWARE WERE EMPLOYED TO REPRESENT ALL THE MAIN COMPONENTS, TAKING INTO ACCOUNT PROJECTILE DEFORMATIONS. THE PENETRATION DEPTH AND DAMAGE AREAS MEASURED WERE THEN COMPARED TO THE EXPERIMENTAL DATA TO VALIDATE THE MODELS.

On solving singular interface problems using the enriched partition‐of‐unity finite element methods

It has been well recognized that interface problems often contain strong singularities which make conventional numerical approaches such as uniform h‐ or p‐version of finite element methods (FEMs) inefficient. In this paper, the partition‐of‐unity finite element method (PUFEM) is applied to obtain solution for interface problems with severe singularities. In the present approach, asymptotical expansions of the analytical solutions near the interface singularities are employed to enhance the accuracy of the solution. Three different enrichment schemes for interface problems are presented, and their performances are studied. Compared to other numerical approaches such as h‐p version of FEM, the main advantages of the present method include: easy and simple formulation; highly flexible enrichment configurations; no special treatment needed for numerical integration and boundary conditions; and highly effective in terms of computational efficiency. Numerical examples are included to illustrate the robustness and performance of the three schemes in conjunction with uniform h‐ or p‐refinements. It shows that the present PUFEM formulations can significantly improve the accuracy of solution. Very often, improved convergence rate is obtained through enrichment in conjunction with p‐refinement.

A Study on the Ricochet of Concrete Debris Against Soil

In this study, the impact responses of concrete debris against soil are investigated. Three types of concrete debris are shot at soil with different incident conditions in experiments. A numerical modeling for the impact process is established and calibrated by the experimental results. A further study on the effect of debris size is then carried out based on the calibrated numerical modeling. A set of formulation is presented to predict the outgoing velocity and the outgoing angle in terms of the incident velocity and the incident angle. Critical lethality curves are derived based on the assumption of a critical kinetic energy of 79 J.

Symmetric BEM formulations for 2-D steady-state and transit potential problems

Abstract This paper presents a new concept for symmetric boundary element method (SBEM) applicable to 2-D steady-state and transit potential problems. Two kinds of SBEM formulations are derived. Symmetry is obtained simply through matrix manipulation, and no hypersingularity appears. Therefore, SBEM is much easier than the traditional symmetric Galerkin BEM. Compared with the traditional asymmetric BEM, the present SBEM can reduce the computational cost for time domain problems only. However, when applied to BEM/FEM coupling procedure, SBEM can reduce the computational cost for both steady-state and time domain problems. Three numerical examples are included to illustrate the effectiveness and accuracy of the present formulations.