A new family of generalized‐α time integration algorithms without overshoot for structural dynamics

A new family of generalized‐α (G‐α) algorithm without overshoot is presented by introducing seven free parameters into the single‐step three‐stage formulation for solution of structural dynamic problems. It is proved through finite difference analysis that these algorithms are unconditionally stable, second‐order accurate and numerical dissipation controllable. The comparison of the new G‐α algorithms with the commonly used G‐α algorithms shows that the newly developed algorithms have the advantage of eliminating the overshooting characteristics exhibited by the commonly used algorithms while their excellent property of dissipation is preserved. The numerical simulation results obtained using a single‐degree‐of‐freedom system and a two‐degree‐of‐freedom system to represent the character of typical large systems coincide well with the results of theoretical analyses. Copyright © 2008 John Wiley & Sons, Ltd.     

The ρ-family of algorithms for time-step integration with improved numerical dissipation

The dynamic analysis of complex non-linear structural systems by the finite element approach requires the use of time-step algorithms for solving the equations of motion in the time domain. Both an implicit and an explicit version of such a time-step algorithm, called the ρ-method, the parameter ρ being used for controlling numerical damping in the higher modes, are presented in this paper. For the implicit family of algorithms unconditional stability, consistency, convergence, accuracy and overshoot properties are first discussed and proved. On the basis of the algorithmic damping ratio (dissipation) and period elongation (dispersion) the ρ-method is then compared with the well-known implicit algorithms of Hilber, Newmark, Wilson, Park and Houbolt. An explicit version of the algorithm is also derived and briefly discussed. This shows numerical properties similar to the central difference method. Both versions of the algorithm have been implemented in a general purpose computer program which has been often used for both numerical tests and practical applications.     

有限差分格式与物理量守恒性

为了成功地开展大气数值模拟,如何构造相应的离散格式是一项十分重要的工作.本文以大气球面浅水方程为例,从不同形式的微分方程组出发,分别构造了能量守恒格式、准辛格式和多守恒格式,并作相应的数值检验和比较,从中可以清楚地看到三类格式的优劣.这三类格式构造方法不同,所保持的物理守恒性不完全相同,数值计算时所需机时也不同,这些比较结果为实际工作的不同使用需求提供了选择依据.     

Improved numerical dissipation for explicit methods in pseudodynamic tests

There is no second-order accurate, dissipative, explicit method in the currently available step-by-step integration algorithms. Two new families of second-order accurate, dissipative, explicit methods have been successfully developed for the direct integration of equations of motion in structural dynamics. These two families of methods are numerically equivalent and possess the desired numerical dissipation which can be continuously controlled. These two families of algorithms are very useful for pseudodynamic tests since the favourable numerical damping can be used to suppress the spurious growth of high-frequency modes due to the presence of numerical and/or experimental errors in performing a pseudodynamic test. © 1997 by John Wiley & Sons, Ltd.     

A numerical laboratory for simulation and visualization of seismic wavefields1

Computational seismic modelling (CSM) plays an important role in the geophysical industry as an established aid to seismic interpreters. Numerical solution of the elastic wave equations has proved to be a very important tool for geophysicists in both forward modelling and migration. Among the techniques generally used in CSM, we consider the finite-element method (FEM) and investigate its computational and visualization requirements. The CSMFEM program, designed for this purpose and developed on an IBM 3090 computer with vector facility, is described in detail. It constitutes a numerical laboratory for performing computer experiments. Two Newmark type algorithms for time integration are compared with other time integration schemes, and both direct and iterative methods for solving the corresponding large sparse system of linear algebraic equations are analysed. Several numerical experiments to simulate seismic energy propagation through heterogeneous media are performed. Synthetics in the form of common shot gathers, vertical seismic profiles and snapshots are suitably displayed, since with the large amounts of data obtained from CSM research, methods for visualization of the computed results must be developed. The FEM is compared with other numerical tools, such as finite-difference and pseudo-spectral methods.     

Two solution methods for dynamic game in reservoir operation

During the last decade, a number of models have been developed to consider the conflict in dynamic reservoir operation. Most of these models are discrete dynamic models which are developed based on game theory. In this study, a continuous model of dynamic game and its corresponding solutions are developed for reservoir operation. Two solution methods are used to solve the model of continuous dynamic game, namely the Ricatti equations and collocation methods. The Ricatti equations method is a closed form solution, requiring less computational efforts compared with discrete models. The collocation solution method applies Newton's method or a quasi-Newton method to find the problem solution. These approaches are able to generate operating policies for dynamic reservoir operation. The Zayandeh-Rud river basin in central Iran is used as a case study and the results are compared with alternative water allocation models. The results show that the proposed solution methods are quite capable of providing appropriate reservoir operating policies, while requiring rather short computational times due to continuous formulation of state and decision variables. Reliability indices are used to compare the overall performance of the proposed models. Based on the results from this study, the collocation method leads to improved values of the reliability indices for total reservoir system and utility satisfaction of water users, compared to the Ricatti equations method. This is attributed to the flexible structure of the collocation model. When compared to alternative water allocation models, lower values of reliability indices are achieved by the collocation method.     

Adaptive volume penalization for ocean modeling

The development of various volume penalization techniques for use in modeling topographical features in the ocean is the focus of this paper. Due to the complicated geometry inherent in ocean boundaries, the stair-step representation used in the majority of current global ocean circulation models causes accuracy and numerical stability problems. Brinkman penalization is the basis for the methods developed here and is a numerical technique used to enforce no-slip boundary conditions through the addition of a term to the governing equations. The second aspect to this proposed approach is that all governing equations are solved on a nonuniform, adaptive grid through the use of the adaptive wavelet collocation method. This method solves the governing equations on temporally and spatially varying meshes, which allows higher effective resolution to be obtained with less computational cost. When penalization methods are coupled with the adaptive wavelet collocation method, the flow near the boundary can be well-resolved. It is especially useful for simulations of boundary currents and tsunamis, where flow near the boundary is important. This paper will give a thorough analysis of these methods applied to the shallow water equations, as well as some preliminary work applying these methods to volume penalization for bathymetry representation for use in either the nonhydrostatic or hydrostatic primitive equations.     

ACTIVE AND SEMI-ACTIVE CONTROL OF STRUCTURES UNDER SEISMIC EXCITATION

Considerable effort has been devoted to develop passive and active methods for reducing structural response under seismic excitations. Passive control approaches have already found application in practice. Active control methods, on the other hand, are being vigorously examined for application to civil structures. This paper investigates the application of active and semi-active control schemes to structures subjected to seismic excitations, and it focuses on the use of the sliding-mode control approach for the development of the control algorithms. The possibility of control redundancy with respect to the number of sliding constraints is taken into account in the controller design. Several sets of numerical results are obtained for a realistic 10-storey shear building, subjected to earthquake-induced ground motions and controlled by active or semi-active control schemes. It is observed that both active and semi-active control schemes can be used to reduce the dynamic response. Active control performs very effectively in reducing the structural response, but the required control force values can be quite large to limit its practical application in the case of large and massive buildings. Active regulation of linear viscous dampers was found unnecessary for this type of structural system, as it did not induce any significantly more reduction in the response than the dampers acting passively. On the other hand, it is shown that active regulation of stiffness can be used with advantage to reduce the response. © 1997 by John Wiley & Sons, Ltd.     

The unsteady plane flow of ice-sheets: A parabolic problem with two moving boundaries

Abstract

Finite difference algorithms have been developed to solve a one-dimensional non-linear parabolic equation with one or two moving boundaries and to analyse the unsteady plane flow of ice-sheets. They are designed to investigate the response of an ice-sheet to changes in climate, and to reconstruct climatic changes implied by past ice-sheet variations inferred from glacial geological data. Two algorithms are presented and compared. The first, a fixed domain method, replaces time as an independent variable with span. The grid interval in real space is kept constant, and thus the number of grid points changes with span. The second, a moving mesh method, retains time as one of the independent variables, but normalises the spatial variable relative to the span, which now enters the diffusion and advection coeficients in the parabolic equation for the surface profile.

Crank-Nicholson schemes for the solution of the equations are constructed, and iterative schemes for the solution of the resulting non-linear equations are considered.

Boundary (margin) motion is governed by the surface slope at the margin. Differentiation of the evolution equations results in an evolution equation for the margin slopes. It is shown that incorporation of this evolution equation, while not formally increasing the accuracy of the finite difference schemes, in practice increases accuracy of the solution.     

Efficient uncertainty quantification techniques in inverse problems for Richards’ equation using coarse-scale simulation models

This paper concerns efficient uncertainty quantification techniques in inverse problems for Richards’ equation which use coarse-scale simulation models. We consider the problem of determining saturated hydraulic conductivity fields conditioned to some integrated response. We use a stochastic parameterization of the saturated hydraulic conductivity and sample using Markov chain Monte Carlo methods (MCMC). The main advantage of the method presented in this paper is the use of multiscale methods within an MCMC method based on Langevin diffusion. Additionally, we discuss techniques to combine multiscale methods with stochastic solution techniques, specifically sparse grid collocation methods. We show that the proposed algorithms dramatically reduce the computational cost associated with traditional Langevin MCMC methods while providing similar sampling performance.     

Limitations on linear multistep methods for structural dynamics

When solving the equations of structural dynamics using direct time integration methods, algorithmic damping is useful to control spurious high-frequency oscillations. Ideally, an algorithm should possess asymptotic annihilation of the high-frequency response, i.e. spurious oscillations are eliminated after one time step. Numerous one-step algorithms, spectrally equivalent to linear multistep (LMS) methods, have been developed which include controlled numerical dissipation. This paper proves that the only unconditionally stable, second-order accurate, 3-step LMS method possessing asymptotic annihilation is Houbolt's method, which is known to be overly dissipative in the low-frequency regime. Thus, using LMS methods, obtaining asymptotic annihilation with little low-frequency dissipation requires at least a 4-step method.     

High-order schemes for 2D unsteady biogeochemical ocean models

Accurate numerical modeling of biogeochemical ocean dynamics is essential for numerous applications, including coastal ecosystem science, environmental management and energy, and climate dynamics. Evaluating computational requirements for such often highly nonlinear and multiscale dynamics is critical. To do so, we complete comprehensive numerical analyses, comparing low- to high-order discretization schemes, both in time and space, employing standard and hybrid discontinuous Galerkin finite element methods, on both straight and new curved elements. Our analyses and syntheses focus on nutrient–phytoplankton–zooplankton dynamics under advection and diffusion within an ocean strait or sill, in an idealized 2D geometry. For the dynamics, we investigate three biological regimes, one with single stable points at all depths and two with stable limit cycles. We also examine interactions that are dominated by the biology, by the advection, or that are balanced. For these regimes and interactions, we study the sensitivity to multiple numerical parameters including quadrature-free and quadrature-based discretizations of the source terms, order of the spatial discretizations of advection and diffusion operators, order of the temporal discretization in explicit schemes, and resolution of the spatial mesh, with and without curved elements. A first finding is that both quadrature-based and quadrature-free discretizations give accurate results in well-resolved regions, but the quadrature-based scheme has smaller errors in under-resolved regions. We show that low-order temporal discretizations allow rapidly growing numerical errors in biological fields. We find that if a spatial discretization (mesh resolution and polynomial degree) does not resolve the solution, oscillations due to discontinuities in tracer fields can be locally significant for both low- and high-order discretizations. When the solution is sufficiently resolved, higher-order schemes on coarser grids perform better (higher accuracy, less dissipative) for the same cost than lower-order scheme on finer grids. This result applies to both passive and reactive tracers and is confirmed by quantitative analyses of truncation errors and smoothness of solution fields. To reduce oscillations in un-resolved regions, we develop a numerical filter that is active only when and where the solution is not smooth locally. Finally, we consider idealized simulations of biological patchiness. Results reveal that higher-order numerical schemes can maintain patches for long-term integrations while lower-order schemes are much too dissipative and cannot, even at very high resolutions. Implications for the use of simulations to better understand biological blooms, patchiness, and other nonlinear reactive dynamics in coastal regions with complex bathymetric features are considerable.     

A comparative study of numerical approaches to risk assessment of contaminant transport

In risk analysis, a complete characterization of the concentration distribution is necessary to determine the probability of exceeding a threshold value. The most popular method for predicting concentration distribution is Monte Carlo simulation, which samples the cumulative distribution function with a large number of repeated operations. In this paper, we first review three most commonly used Monte Carlo (MC) techniques: the standard Monte Carlo, Latin Hypercube sampling, and Quasi Monte Carlo. The performance of these three MC approaches is investigated. We then apply stochastic collocation method (SCM) to risk assessment. Unlike the MC simulations, the SCM does not require a large number of simulations of flow and solute equations. In particular, the sparse grid collocation method and probabilistic collocation method are employed to represent the concentration in terms of polynomials and unknown coefficients. The sparse grid collocation method takes advantage of Lagrange interpolation polynomials while the probabilistic collocation method relies on polynomials chaos expansions. In both methods, the stochastic equations are reduced to a system of decoupled equations, which can be solved with existing solvers and whose results are used to obtain the expansion coefficients. Then the cumulative distribution function is obtained by sampling the approximate polynomials. Our synthetic examples show that among the MC methods, the Quasi Monte Carlo gives the smallest variance for the predicted threshold probability due to its superior convergence property and that the stochastic collocation method is an accurate and efficient alternative to MC simulations.     

Lax-Wendroff and Nyström methods for seismic modelling

Lax-Wendroff and Nyström methods are numerical algorithms of temporal approximations for solving differential equations. These methods provide efficient algorithms for high-accuracy seismic modeling. In the context of spatial pseudospectral discretizations, I explore these two kinds of methods in a comparative way. Their stability and dispersion relation are discussed in detail. Comparison between the fourth-order Lax-Wendroff method and a fourth-order Nyström method shows that the Nyström method has smaller stability limit but has a better dispersion relation, which is closer to the sixth-order Lax-Wendroff method. The structure-preserving property of these methods is also revealed. The Lax-Wendroff methods are a second-order symplectic algorithm, which is independent of the order of the methods. This result is useful for understanding the error growth of Lax-Wendroff methods. Numerical experiments based on the scalar wave equation are performed to test the presented schemes and demonstrate the advantages of the symplectic methods over the nonsymplectic ones.     

Development of a family of unconditionally stable explicit direct integration algorithms with controllable numerical energy dissipation

The implicit dissipative generalized‐ α method is analyzed using discrete control theory. Based on this analysis, a one‐parameter family of explicit direct integration algorithms with controllable numerical energy dissipation, referred to as the explicit KR‐α method, is developed for linear and nonlinear structural dynamic numerical analysis applications. Stability, numerical dispersion, and energy dissipation characteristics of the proposed algorithms are studied. It is shown that the algorithms are unconditionally stable for linear elastic and stiffness softening‐type nonlinear systems, where the latter indicates a reduction in post yield stiffness in the force–deformation response. The amount of numerical damping is controlled by a single parameter, which provides a measure of the numerical energy dissipation at higher frequencies. Thus, for a specific value of this parameter, the resulting algorithm is shown to produce no numerical energy dissipation. Furthermore, it is shown that the influence of the numerical damping on the lower mode response is negligible. It is further shown that the numerical dispersion and energy dissipation characteristics of the proposed explicit algorithms are the same as that of the implicit generalized‐ α method. A numerical example is presented to demonstrate the potential of the proposed algorithms in reducing participation of undesired higher modes by using numerical energy dissipation to damp out these modes. Copyright © 2014 John Wiley & Sons, Ltd.     

Point Mass Method Applied to the Regional Gravimetric Determination of the Geoid

The determination of the local gravity field by means of the point mass inversion method can be performed as an alternative to conventional numerical methods, such as the least-squares collocation. Based on the first derivative of the inverse-distance Newtonian potential for the representation of the gravity anomaly data, it is possible to compute any wavelength component of the geoid in planar approximation with sufficient accuracy. In order to exemplify the theoretical concept, two applications are presented of the computation of two different wavelength components of the geoid, the long wavelength component in a local solution and the short wavelength component in a regional solution. The results are compared with corresponding least-squares collocation solutions, using a global geopotential model to remove and to restore the long wavelength component.     

Complex-parameter kelvin model for elastic foundations

A new rigorous approach in modelling mechanical behaviour is developed. The dynamic response of a rigid disc resting on an elastic half-space is approximated by a macroscopic force-displacement relationship, where not only the coefficients but also the order of time derivatives are complex valued. It is shown that comlex-parameter constitutive models are not simply an elegant alternative in modelling mechanical behaviour, but are the outcome of rigorous non-linear regression analysis on complex-valued functions, like the dynamic stiffness of dissipative systems. Complex-parameter constitutive models are very attractive since a minimum number of parameters is required to obtain a satisfactory fit of the ‘exact’ response. A two-parameter generalized Kelvin model reproduces closely the ‘exact’ dynamic stiffness of the disc for all three vertical, horizontal and rocking modes studied herein. Frequency- and time-domain algorithms to solve complex-order differential equations are developed, validated and used to calculate the foundation–response under earthquake excitation. It is the excellent agreement of results and the economy in number of parameters that make complex-parameter models so attractive in constitutive modelling.     

Fast modelling of marine surface multiples

In the development and testing of water‐surface multiple‐removal algorithms, it is valuable to have accurate synthetic seismograms which exhibit multiples, for which the multiple‐free solution is known. A method is presented for constructing 2D and 3D solutions of the acoustic wave equation in water, by combining the solution from a primary source with other scaled solutions of secondary sources, which simulate diffractors. The computation involves function evaluation rather than numerical solution of differential equations and is consequently accurate and comparatively fast. The analytic formulae on which the method is based give insights into methods for multiple removal. Generalized reflection coefficients, defined on a horizontal plane above the diffractors, are derived and used to construct the integral equations which are the basis for many multiple‐removal schemes.     

EVALUATION OF SOME IMPLICIT TIME-STEPPING ALGORITHMS FOR PSEUDODYNAMIC TESTS

Two types of implicit time-stepping algorithms have been proposed recently for pseudodynamic tests. The first type consists of an algorithm which relies on Newton iterations to satisfy the equations of motion. The second type consists of an algorithm which is based on the Operator-Splitting technique and does not require any numerical iteration. While one or the other has been preferred by some researchers, these time-stepping algorithms have not been analysed and compared under a uniform setting. In this paper, a concise summary of these schemes is presented, and they are evaluated in a consistent manner in terms of numerical dissipation, frequency distortion and experimental errors. The analytical results are validated by numerical simulations as well as experimental results. It is shown that the algorithm based on Newton iterations can control experimental error effects effectively by means of an error-correction procedure. The algorithm based on the Operator-Splitting technique demonstrates similar performance provided the I-Modification is adopted.     

Direct BEM for high-resolution global gravity field modelling

The paper presents a high-resolution global gravity field modelling by the boundary element method (BEM). A direct BEM formulation for the Laplace equation is applied to get a numerical solution of the linearized fixed gravimetric boundary-value problem. The numerical scheme uses the collocation method with linear basis functions. It involves a discretization of the complicated Earth’s surface, which is considered as a fixed boundary. Here 3D positions of collocation points are simulated from the DNSC08 mean sea surface at oceans and from the SRTM30PLUS_V5.0 global topography model added to EGM96 on lands. High-performance computations together with an elimination of the far zones’ interactions allow a very refined integration over the all Earth’s surface with a resolution up to 0.1 deg. Inaccuracy of the approximate coarse solutions used for the elimination of the far zones’ interactions leads to a long-wavelength error surface included in the obtained numerical solution. This paper introduces an iterative procedure how to reduce such long-wavelength error surface. Surface gravity disturbances as oblique derivative boundary conditions are generated from the EGM2008 geopotential model. Numerical experiments demonstrate how the iterative procedure tends to the final numerical solutions that are converging to EGM2008. Finally the input surface gravity disturbances at oceans are replaced by real data obtained from the DNSC08 altimetryderived gravity data. The ITG-GRACE03S satellite geopotential model up to degree 180 is used to eliminate far zones’ interactions. The final high-resolution global gravity field model with the resolution 0.1 deg is compared with EGM2008.