Adaptive damping for dynamic relaxation problems with non‐monotonic spectral response

Non‐inertial transients may be effectively solved using explicit time integration with arbitrary inertial and damping properties. The usual approach relies on a diagonal lumped mass on which strictly mass proportional damping is based. While the damping coefficient can be adaptively determined on the basis of the estimated frequency of the predominant response, local changes in stiffness, often associated with changing contact conditions, can cause abrupt changes in the damping coefficient. This can substantially impede the progress of the solution, particularly when deformations involve large translations or rotations of parts of the system. A modification to the mass proportional damping is developed and implemented to avoid the deleterious effects of sudden changes in the damping coefficient. A new procedure is thus implemented so that the usual dynamic relaxation method will automatically adapt to changing conditions of response in such a way as to avoid overdamping low modes due to subsequent higher frequency events. Copyright © 2002 John Wiley & Sons, Ltd.     

Inverse problems for estimating bottom boundary conditions of natural waters

Unknown boundary conditions for natural waters are estimated using an inverse problem methodology. In the formulation of the inverse problem, expressed as a non‐linear constrained optimization problem, its objective function is given by a square difference term, between experimental and computed data, added to a regularization operator. The computed data are obtained by solving the radiative transfer direct problem using the LTS_N method. A key aspect to get a good reconstruction is played by observed data quality. The reconstruction strategy is examined for in situ radiance and irradiance data for many arrangements of the experimental grid of the measurement devices, in order to plan good designs for experimental works. Copyright © 2002 John Wiley & Sons, Ltd.     

Convergence analysis of a parallel interfield method for heterogeneous simulations with dynamic substructuring

In order to predict the dynamic response of a complex system decomposed by computational or physical considerations, partitioned procedures of coupled dynamical systems are needed. This paper presents the convergence analysis of a novel parallel interfield procedure for time‐integrating heterogeneous (numerical/physical) subsystems typical of hardware‐in‐the‐loop and pseudo‐dynamic tests. The partitioned method is an extension of the method originally proposed by Gravouil and Combescure which utilizes a domain decomposition enforcing the continuity of the velocity at interfaces. In particular, the merits of the new method which can couple arbitrary Newmark schemes with different time steps in different subdomains and advance all the substructure states simultaneously are analysed in terms of accuracy and stability. All theoretical results are derived for single‐ and two‐degrees‐of‐freedom systems, as a multi‐degree‐of‐freedom system is too difficult to analyse mathematically. However, the insight gained from the analysis of these coupled problems and the conclusions drawn are confirmed by means of the numerical simulation on a four‐degrees‐of‐freedom system. Copyright © 2008 John Wiley & Sons, Ltd.     

The probability density evolution method for dynamic response analysis of non‐linear stochastic structures

The probability density evolution method (PDEM) for dynamic responses analysis of non‐linear stochastic structures is proposed. In the method, the dynamic response of non‐linear stochastic structures is firstly expressed in a formal solution, which is a function of the random parameters. In this sense, the dynamic responses are mutually uncoupled. A state equation is then constructed in the augmented state space. Based on the principle of preservation of probability, a one‐dimensional partial differential equation in terms of the joint probability density function is set up. The numerical solving algorithm, where the Newmark‐Beta time‐integration algorithm and the finite difference method with Lax–Wendroff difference scheme are brought together, is studied. In the numerical examples, free vibration of a single‐degree‐of‐freedom non‐linear conservative system and dynamic responses of an 8‐storey shear structure with bilinear hysteretic restoring forces, subjected to harmonic excitation and seismic excitation, respectively, are investigated. The investigations indicate that the probability density functions of dynamic responses of non‐linear stochastic structures are usually irregular and far from the well‐known distribution types. They exhibit obvious evolution characteristics. The comparisons with the analytical solution and Monte Carlo simulation method demonstrate that the proposed PDEM is of fair accuracy and efficiency. Copyright © 2005 John Wiley & Sons, Ltd.     

The method of fundamental solutions for problems in static thermo-elasticity with incomplete boundary data

An inverse problem in static thermo-elasticity is investigated. The aim is to reconstruct the unspecified boundary data, as well as the temperature and displacement inside a body from over-specified boundary data measured on an accessible portion of its boundary. The problem is linear but ill-posed. The uniqueness of the solution is established but the continuous dependence on the input data is violated. In order to reconstruct a stable and accurate solution, the method of fundamental solutions is combined with Tikhonov regularization where the regularization parameter is selected based on the L-curve criterion. Numerical results are presented in both two and three dimensions showing the feasibility and ease of implementation of the proposed technique.     

Efficient dynamic finite element method for 3D objects with uniform cross‐section

An efficient implicit dynamic finite element method (FEM) for elastic 3D objects with uniform cross‐sections was developed. In this method, the finite element mesh is generated in such a way that the object to be analysed is at first sliced into layers with the same thickness along its generatrix and then each layer is discretized into finite elements of the same pattern. This way of discretization makes the mass, viscosity, and stiffness matrices into the repetitive block tridiagonal matrices. The repetitive block tridiagonal matrix has the characteristic, that the sequence of matrices which appears in the Gaussian elimination for the repetitive block tridiagonal matrix is a rapid convergent sequence. The process of the Gaussian elimination can be terminated when the sequence converges. The rest of the sequence is not necessary to be stored. The present method can save the computational time and memory by utilising this characteristic of the repetitive block tridiagonal matrix. A few examples of analyses including whole Hopkinson‐bar analysis were performed to demonstrate the effectiveness of the present method. The present method is applicable not only to the elasto‐dynamics but also to many other problems, such as thermal problems, electrical problems, and plastic problems without geometric non‐linearity. Copyright © 2005 John Wiley & Sons, Ltd.     

Boundary knot method for some inverse problems associated with the Helmholtz equation

The boundary knot method is an inherently meshless, integration‐free, boundary‐type, radial basis function collocation technique for the solution of partial differential equations. In this paper, the method is applied to the solution of some inverse problems for the Helmholtz equation, including the highly ill‐posed Cauchy problem. Since the resulting matrix equation is badly ill‐conditioned, a regularized solution is obtained by employing truncated singular value decomposition, while the regularization parameter for the regularization method is provided by the L‐curve method. Numerical results are presented for both smooth and piecewise smooth geometry. The stability of the method with respect to the noise in the data is investigated by using simulated noisy data. The results show that the method is highly accurate, computationally efficient and stable, and can be a competitive alternative to existing methods for the numerical solution of the problems. Copyright © 2005 John Wiley & Sons, Ltd.     

The bi-potential method applied to the modeling of dynamic problems with friction

The bi-potential method has been successfully applied to the modeling of frictional contact problems in static cases. This paper presents an extension of this method for dynamic analysis of impact problems with deformable bodies. A first order algorithm is applied to the numerical integration of the time-discretized equation of motion. Using the Object-Oriented Programming (OOP) techniques in C++ and OpenGL graphical support, a finite element code including pre/postprocessor FER/Impact is developed. The numerical results show that, at the present stage of development, this approach is robust and efficient in terms of numerical stability and precision compared with the penalty method.     

A time‐dependent formulation of dual boundary element method for 3D dynamic crack problems

In this paper, the dual boundary element method in time domain is developed for three‐dimensional dynamic crack problems. The boundary integral equations for displacement and traction in time domain are presented. By using the displacement equation and traction equation on crack surfaces, the discontinuity displacement on the crack can be determined. The integral equations are solved numerically by a time‐stepping technique with quadratic boundary elements. The dynamic stress intensity factors are calculated from the crack opening displacement. Several examples are presented to demonstrate the accuracy of this method. Copyright © 1999 John Wiley & Sons, Ltd     

The method of fundamental solutions for inverse source problems associated with the steady‐state heat conduction

This paper presents the use of the method of fundamental solutions (MFS) for recovering the heat source in steady‐state heat conduction problems from boundary temperature and heat flux measurements. It is well known that boundary data alone do not determine uniquely a general heat source and hence some a priori knowledge is assumed in order to guarantee the uniqueness of the solution. In the present study, the heat source is assumed to satisfy a second‐order partial differential equation on a physical basis, thereby transforming the problem into a fourth‐order partial differential equation, which can be conveniently solved using the MFS. Since the matrix arising from the MFS discretization is severely ill‐conditioned, a regularized solution is obtained by employing the truncated singular value decomposition, whilst the optimal regularization parameter is determined by the L‐curve criterion. Numerical results are presented for several two‐dimensional problems with both exact and noisy data. The sensitivity analysis with respect to two solution parameters, i.e. the number of source points and the distance between the fictitious and physical boundaries, and one problem parameter, i.e. the measure of the accessible part of the boundary, is also performed. The stability of the scheme with respect to the amount of noise added into the data is analysed. The numerical results obtained show that the proposed numerical algorithm is accurate, convergent, stable and computationally efficient for solving inverse source problems in steady‐state heat conduction. Copyright © 2006 John Wiley & Sons, Ltd.     

Imposition of boundary conditions by modifying the weighting coefficient matrices in the differential quadrature method

One of the important issues in the implementation of the differential quadrature method is the imposition of the given boundary conditions. There may be multiple boundary conditions involving higher‐order derivatives at the boundary points. The boundary conditions can be imposed by modifying the weighting coefficient matrices directly. However, the existing method is not robust and is known to have many limitations. In this paper, a systematic procedure is proposed to construct the modified weighting coefficient matrices to overcome these limitations. The given boundary conditions are imposed exactly. Furthermore, it is found that the numerical results depend only on those sampling grid points where the differential quadrature analogous equations of the governing differential equations are established. The other sampling grid points with no associated boundary conditions are not essential. Copyright © 2002 John Wiley & Sons, Ltd.     

Wave finite element‐based superelements for forced response analysis of coupled systems via dynamic substructuring

The wave finite element (WFE) method is used for assessing the harmonic response of coupled mechanical systems that involve one‐dimensional periodic structures and coupling elastic junctions. The periodic structures under concern are composed of complex heterogeneous substructures like those encountered in real engineering applications. A strategy is proposed that uses the concept of numerical wave modes to express the dynamic stiffness matrix (DSM), or the receptance matrix (RM), of each periodic structure. Also, the Craig–Bampton (CB) method is used to model each coupling junction by means of static modes and fixed‐interface modes. An efficient WFE‐based criterion is considered to select the junction modes that are of primary importance. The consideration of several periodic structures and coupling junctions is achieved through classic finite element (FE) assembly procedures, or domain decomposition techniques. Numerical experiments are carried out to highlight the relevance of the WFE‐based DSM and RM approaches in terms of accuracy and computational savings, in comparison with the conventional FE and CB methods. The following test cases are considered: a 2D frame structure under plane stresses and a 3D aircraft fuselage‐like structure involving stiffened cylindrical shells. Copyright © 2015 John Wiley & Sons, Ltd.     

Equilibrium on line method (ELM) for imposition of Neumann boundary conditions in the finite point method (FPM)

Equilibrium on line method (ELM) for imposition of Neumann boundary conditions in the finite point method (FPM) is presented. In contrary to weak‐form‐based methods, strong‐form‐based methods such as the FPM are often unstable and less accurate, especially for problems governed by partial differential equations with Neumann (derivative) boundary conditions. In this paper, a truly meshless approach for imposition of Neumann boundary conditions in the FPM is proposed and adopted for 2D elasticity analyses. In the proposed method, equilibrium on lines on the Neumann boundary conditions is satisfied as Neumann boundary condition equations. Numerical studies show that this method for imposition of Neumann boundary is simple to implement and computationally efficient and also leads to more stable and accurate results. Copyright © 2006 John Wiley & Sons, Ltd.     

The mixed finite element method for the quasi‐static and dynamic analysis of viscoelastic timoshenko beams

The quasi‐static and dynamic responses of a linear viscoelastic Timoshenko beam on Winkler foundation are studied numerically by using the hybrid Laplace–Carson and finite element method. In this analysis the field equation for viscoelastic material is used. In the transformed Laplace–Carson space two new functionals have been constructed for viscoelastic Timoshenko beams through a systematic procedure based on the Gâteaux differential. These functionals have six and two independent variables respectively. Two mixed finite element formulations are obtained; TB12 and TB4. For the inverse transform Schapery and Fourier methods are used. The numerical results for quasi‐static and dynamic responses of several visco‐elastic models are presented. Copyright © 1999 John Wiley & Sons, Ltd.     

Numerical manifold method for dynamic consolidation of saturated porous media with three-field formulation

In terms of the three-field formulation of Biot's dynamic consolidation theory, the numerical manifold method (NMM) is developed, where the same approximation to skeleton displacement ( u ) and fluid velocity ( w ) is employed and able to reflect incompressible as well as compressible deformation, but the approximation to pore pressure (p ) takes two different types, respectively. The first type of approximation to p is continuous piecewise linear interpolation and the second type assumes that p is a constant within each element. It is verified that using the second type of approximation to p naturally satisfies the inf-sup condition even in the limits of rigid skeleton and very low permeability, avoiding the locking problem accordingly. Energy components done by various forces are calculated to verify the accuracy and stability of the time integration scheme. A mass lumping technique in the NMM framework is employed to effectively reduce the unphysical oscillations and increase computational efficiency, which is another unique advantage of NMM over other numerical methods. A number of numerical tests are conducted to demonstrate the robustness and versatility of the proposed mixed NMM models.     

Interval finite element method for dynamic response of closed‐loop system with uncertain parameters

In practical engineering, it is difficult to obtain all possible solutions of dynamic responses with sharp bounds even if an optimum scheme is adopted where there are many uncertain parameters. In this paper, using the interval finite element (IFE) method and precise time integration (PTI) method, we discuss the dynamic response of vibration control problem of structures with interval parameters. With matrix perturbation theory and interval arithmetic, the algorithm for estimating upper and lower bounds of dynamic response of the closed‐loop system is developed directly from the interval parameters. Two numerical examples are given to illustrate the application of the present method. The example 1 is used to show the applicability of the present method. The example 2 is used to show the validity of the present method by comparing the results with those obtained by the classical random perturbation method. Copyright © 2006 John Wiley & Sons, Ltd.     

Probability density evolution method for dynamic response analysis of structures with uncertain parameters

Probability density evolution method is proposed for dynamic response analysis of structures with random parameters. In the present paper, a probability density evolution equation (PDEE) is derived according to the principle of preservation of probability. With the state equation expression, the PDEE is further reduced to a one-dimensional partial differential equation. The numerical algorithm is studied through combining the precise time integration method and the finite difference method with TVD schemes. The proposed method can provide the probability density function (PDF) and its evolution, rather than the second-order statistical quantities, of the stochastic responses. Numerical examples, including a SDOF system and an 8-story frame, are investigated. The results demonstrate that the proposed method is of high accuracy and efficiency. Some characteristics of the PDF and its evolution of the stochastic responses are observed. The PDFs evidence heavy variance against time. Usually, they are much irregular and far from well-known regular distribution types. Additionally, the coefficients of variation of the random parameters have significant influence on PDF and second-order statistical quantities of responses of the stochastic structure.The support of the Natural Science Funds for Distinguished Young Scholars of China (Grant No.59825105) and the Natural Science Funds for Innovative Research Groups of China (Grant No.50321803) are gratefully appreciated.