Time Integration Algorithm Research Articles

Pseudo-dynamic analysis of a cemented hip arthroplasty using a force method based on the Newmark algorithm

The main goal of this work was to develop an approached model to study dynamic behavior and prediction of the stress distribution in an in vitro Charnley cemented hip arthroplasty. An alternative version of the described pseudo-dynamic procedure is proposed by using the time integration Newmark algorithm. An internal restoring force vector is numerically calculated from the displacement, velocity, and acceleration vectors. A numerical model of hip replacement was developed to analyze the deformation of a dynamically stressed structure for all time steps. The experimental measurement of resulting internal forces generated in the structure (internal restoring force vector) is the second fundamental step of the pseudo-dynamic procedure. These data (as a feedback) are used by the time integration algorithm, which allows updating of the structure's shape for the next displacement, velocity, and acceleration vectors. In the field of Biomechanics, the potentialities of this method contribute to the determination of a dynamically equivalent in vitro stress field of a cemented hip prosthesis; implant fitted in patients with a normal mobility or practice sports. Consequences of the stress distribution in the implant zone that underwent cyclic fatigue loads were also discussed by using a finite element model. Application of this method in Biomechanics appears as a useful tool in the approximate stress field characterization of the peak stress state. Results show a peak value around two times the static situation, more for making possible the prediction of future damage and a programed clinical examination in patients using hip prosthesis.

Enhanced low‐order displacement finite elements using incompatible mass scaling and static condensation in explicit time integration

AbstractFor nonlinear transient problems as well as rather slow quasi‐static processes explicit time integration has proven to be a suitable and robust time integration algorithm. As a consequence of the Courant‐Friedrichs‐Lewy (CFL) criterion small time step sizes are necessary and the overall computation time is dominated by the operations on element level. Privileged use of low order finite elements is common in explicit time integration due to their efficiency and robustness. In order to suppress artificial locking effects the work of Bischoff and Romero [1] is followed where a generalization of the method of incompatible modes (IM) is derived being equivalent to the class of enhanced assumed strain (EAS) elements originally proposed by Simo and Rifai [2]. These concepts are bound to a static elimination procedure of internal parameters which may result in a considerable expansion of the computational costs in the case of an explicit time integration scheme. In order to avoid the static elimination procedure inertia is considered for the incompatible parameters as suggested by Mattern et al. [3, 4]. This step allows integrating the incompatible parameters directly in time, but since rather large stiffnesses are related to the incompatible parameters – particularly for the volumetric mode enhancements – the highest eigenfrequency may be increased which would lead to a decrease of the time step size. This motivates scaling the masses associated to these incompatible parameters to control their effect on the time step size. It is possible to derive analytically based estimates for the scaling of the IM‐masses in order to achieve results agreeing very good with the classical EAS approach including large deflections and nonlinear material behaviour with elasto‐plasticity. The numerical implementation is performed using AceGen [5]. (© 2014 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Corotational Nonlinear Dynamic Analysis of Thin-Shell Structures with Finite Rotations

A new formulation for corotational nonlinear dynamic analysis of thin-shell structures involving large displacements and finite rotations has been presented in this paper. By introducing the kinematic description of the geometrically exact quasi-static shell model to describe the motion of an arbitrary material point in a triangular shell element, the new expressions for the inertial force vector and tangent inertia matrix were derived systematically. The elastic deformation was handled by using the consistent symmetrizable equilibrated corotational formulation, which is independent with respect to the local finite element modeling. An energy-momentum-conserving algorithm and an energy-decaying/momentum-conserving algorithm were developed for nonlinear dynamic analysis by applying the generalized energy-momentum method in the corotational formulation. Three classic numerical examples, including the dynamic response, free motion, and dynamic buckling of thin-shell structures, were computed to verify the accuracy and capability of the presented formulation for solving the nonlinear dynamic problems of thin-shell structures with finite rotations. In the relevant numerical example, the conservations of linear momentum and angular momentum of a shell structure due to the time-integration algorithms in the corotational formulation were investigated.

Testing and numerical simulation of Holocene marine sand uniaxial compression along the Lithuanian coast

Compressibility of quartz sand from the Lithuanian coastal area in Klaipėda environs is investigated by testing and numerical simulation, with validation of obtained results. The shape of sand grains has been analysed with a scanning electronic microscope (SEM). The determined morphological parameters of sand grains are employed to create discrete models (particle models of grains) subsequently used for sand compression test numerical simulation via discrete element method (DEM) techniques. The background version of DEM and the numerical time-integration algorithm are implemented in original DEMMAT code. Compression tests have been realised by an oedometer device. Test versus numerical simulation results have revealed a dependence of significant compression curve character on the discretised shape of sand grains and Young’s modulus of particles.

An explicit time integration scheme of numerical manifold method

The traditional numerical manifold method (NMM) has the advantage of simulating a continuum and a discontinuum in a unified framework based on a dual cover system. However, since an implicit time integration algorithm is used, the computational efficiency of the original NMM is very low, especially when more contacts are involved. The present study proposes an explicit version of the NMM. Since a lumped mass matrix is used for the manifold element, the accelerations by the corresponding physical covers can be solved explicitly without forming a global stiffness matrix. The open–close iteration is still applied to ensure computational accuracy. The developed method is first validated by two examples, and a highly fractured rock slope is subsequently simulated. Results show that the computational efficiency of the proposed explicit NMM has been significantly improved without losing the accuracy. The explicit NMM is more suitable for large-scale rock mass stability analysis and it deserves to be further developed for engineering computations in rock engineering.

Acoustics in anisotropic poroelasticity: space–time variational formulation

A variational space–time formulation to treat acoustics problems in porous medium is developed. At first, no separation of the space and time has been applied in the approximation of the field variables. This produces a matrix variational formulation for more general numerical treatments. Finally, separation of variables has been applied to produce a simpler space–time formulation, in which the numerical treatment of the space domain and the time integration algorithms can be identified. The numerical treatment has been oriented to find symmetry in the space and time domains. Numerical simulation in the frequency domain and in the time domain have been performed to validate the variational formulation with separation of variables.

A mortar approach for Fluid–Structure interaction problems: Immersed strategies for deformable and rigid bodies

This paper proposes a new Fluid–Structure Interaction immersed computational framework. The coupling between an underlying incompressible fluid and an embedded solid is formulated by means of the overlapping domain decomposition method in conjunction with a mortar approach, leading to a variationally consistent scheme which is capable of unifying a range of methodologies currently available in the literature. This novel framework provides great flexibility and enables the modelling of immersed deformable solids (compressible and incompressible) as well as rigid bodies through the use of a weak director based formulation. A novel Null-Space reduction scheme is employed in order to enhance the conditioning of the resulting system of equations and reduce the computational cost. An implicit structure preserving time integration algorithm is used to yield extra stability and robustness and the use of a segmentation technique near the boundary between fluid and solid also leads to enhanced accuracy. The methodology is benchmarked against results obtained by using alternative boundary fitted methodologies.

Efficiency and Accuracy Verification of the Explicit Numerical Manifold Method for Dynamic Problems

The original numerical manifold method (NMM) employs an implicit time integration scheme to achieve higher computational accuracy, but its efficiency is relatively low, especially when the open–close iterations of contact are involved. To improve its computational efficiency, a modified version of the NMM based on an explicit time integration algorithm is proposed in this study. The lumped mass matrix, internal force and damping vectors are derived for the proposed explicit scheme. A calibration study on P-wave propagation along a rock bar is conducted to investigate the efficiency and accuracy of the developed explicit numerical manifold method (ENMM) for wave propagation problems. Various considerations in the numerical simulations are discussed, and parametric studies are carried out to obtain an insight into the influencing factors on the efficiency and accuracy of wave propagation. To further verify the capability of the proposed ENMM, dynamic stability assessment for a fractured rock slope under seismic effect is analysed. It is shown that, compared to the original NMM, the computational efficiency of the proposed ENMM can be significantly improved.

Finite Element Simulation of Diffusion and Precipitation with Application to Internal Oxidation of Ni-Xwt%Cr Alloys at 950°C

For the simulation of internal oxidation phenomena, different numerical approaches are proposed in the literature based on 1D finite differences or on explicit time integration schemes which need small time-steps leading to very long computation times. The aim of this paper is to detail a multi-dimentional finite element approach which is coupled with an efficient implicit time integration algorithm. The thermodynamic activities and the total mass fractions are both used as principal nodal variables. The use of finite elements rather than finite differences greatly facilitates the meshing of 2D and 3D bodies. Its implicit time-integration allows using much larger time-steps without any degradation of the results. An application is proposed for the modeling of internal oxidation of chromia for Ni-Xwt%Cr alloys at 950°C by considering the barrier effect of precipitates.

Numerical analysis of Lamb waves using the finite and spectral cell methods

An accurate and efficient simulation of wave propagation phenomena plays an important role in different engineering disciplines. In structural health monitoring, for example, ultrasonic guided waves are used to detect and localize damage and to assess the structural integrity of the component part under consideration. Because of the complexity of real structures, the numerical simulation of structural health monitoring systems is a computationally demanding task. Therefore, to facilitate the analysis of wave propagation phenomena, the authors propose to combine the finite cell method with the spectral element method. The ensuing novel method is referred to as the spectral cell method. Because it does not rely on body-fitted meshes, the resulting approach eliminates all discretization difficulties encountered in conventional finite element methods. Moreover, with the aid of mass lumping, it paves the way for the use of explicit time-integration algorithms. In the first part of the paper, we show that using a lumped mass matrix instead of the consistent one has no detrimental effect on the accuracy of the spectral element method. We introduce the spectral cell method in the second part, showing that, when applied to wave propagation analysis, the spectral cell method yields results comparable with other standard higher order finite element approaches.Copyright © 2014 John Wiley & Sons, Ltd.

Finite and spectral cell method for wave propagation in heterogeneous materials

In the current paper we present a fast, reliable technique for simulating wave propagation in complex structures made of heterogeneous materials. The proposed approach, the spectral cell method, is a combination of the finite cell method and the spectral element method that significantly lowers preprocessing and computational expenditure. The spectral cell method takes advantage of explicit time-integration schemes coupled with a diagonal mass matrix to reduce the time spent on solving the equation system. By employing a fictitious domain approach, this method also helps to eliminate some of the difficulties associated with mesh generation. Besides introducing a proper, specific mass lumping technique, we also study the performance of the low-order and high-order versions of this approach based on several numerical examples. Our results show that the high-order version of the spectral cell method together requires less memory storage and less CPU time than other possible versions, when combined simultaneously with explicit time-integration algorithms. Moreover, as the implementation of the proposed method in available finite element programs is straightforward, these properties turn the method into a viable tool for practical applications such as structural health monitoring [1---3], quantitative ultrasound applications [4], or the active control of vibrations and noise [5, 6].

Nonlinear higher-order transient solver for magnetic fluid hyperthermia

The article is devoted to numerical methods for transient solution of Pennes’ bioheat equation as required for Magnetic Fluid Hyperthermia (MFH) modeling. Special attention has been paid to the role of non-linearity of blood perfusion and its influence on temperature distribution. The authors show that the higher-order time integration algorithms are highly advised for this type of problem, which should be classified as a stiff one. Popular low-order solvers give very different solutions. Furthermore, the application of adaptive time stepping scheme reduces calculation time and raises the efficiency of the simulation software.

A Combined Approach of RIBEM and Precise Time Integration Algorithm for Solving Transient Heat Conduction Problems

A combined approach of the radial integration boundary-element method (RIBEM) and the precise algorithm in the time domain is presented for solving transient heat conduction problems with heat sources. First, by expanding physical quantities at discrete time intervals, the recursive formulations of the governing equation are derived. Then, the recursive equation is solved by the RIBEM. Finally, a self-adaptive check technique is carried out to estimate how many expansion terms are needed in a time-step size. Numerical results show that the present approach can obtain satisfactory performance.

Computer Implementation of the DRBEM for Studying the Generalized Thermo Elastic Responses of Functionally Graded Anisotropic Rotating Plates with Two Relaxation Times

A numerical computer model based on the dual reciprocity boundary element method (DRBEM) is extended to study the generalized thermoelastic responses of functionally graded anisotropic rotating plates. In the case of plane deformation, a predictor-corrector implicit-explicit time integration algorithm was developed and implemented for use with the DRBEM to obtain the solution for the displacement and temperature fields in the context of the Lord and Shulman theory. Numerical results that demonstrate the validity of the proposed method are also presented in the tables. 2010 Mathematics subject classification: 74B05 74E05 74F05 74H05 74S20

Computer Implementation of the Drbem for Studying the Classical Coupled Thermoelastic Responses of Functionally Graded Anisotropic Plates

A numerical computer model based on the dual reciprocity boundary element method (DRBEM) is extended to study the generalized thermoelastic responses of functionally graded anisotropic rotating plates. In the case of plane deformation, a predictor-corrector implicit-explicit time integration algorithm was developed and implemented for use with the DRBEM to obtain the solution for the displacement and temperature fields in the context of the classical coupled thermo-elasticity theory. Numerical results that demonstrate the validity of the proposed method are also presented in the tables.

SUPERSONIC FLUTTER CONTROL OF AN ELECTRORHEOLOGICAL FLUID-BASED SMART CIRCULAR CYLINDRICAL SHELL

In this paper, active flutter suppression of a simply supported circular sandwich cylindrical shell with a tunable electrorheological fluid (ERF) core, under axial supersonic gas flow, is studied. The structural analysis is based on the classical thin shell theory, the ERF core is modeled as a first-order Kelvin–Voigt material, and the Krumhaar's modified supersonic piston theory is utilized to model the aerodynamic loading. Hamilton's principle is used to formulate the dynamic equations of motion together with the relevant boundary conditions. The generalized Fourier expansions in the circumferential and axial directions in conjunction with the classical Galerkin method are employed to set up the governing equations in the state-space domain. The critical free stream static pressures at which unstable oscillations arise are calculated for selected applied electric field strengths and cylinder length ratios. The Runge–Kutta time integration algorithm is used to determine the open-loop aeroelastic response of the system in two basic loading configurations, namely, a concentrated impulse point load and a sonic boom line load. Subsequently, a sliding mode control (SMC) strategy is adopted to actively suppress the closed loop system dynamic response in supersonic flight condition. Simulation results demonstrate performance and effectiveness of the adopted ERF-based SMC scheme. Limiting cases are considered and good agreements with the data available in the literature are obtained.

Effect of Twist and Striker Velocity of Torsion Stiff Composite Conical Shells under Multiple Delaminations

The low velocity impact response of torsion stiff composite pretwisted conical shells is portrayed in the present study. The formulation is based on Mindlin's theory incorporating rotary inertia and effects of transverse shear deformation. An eight noded isoparametric plate bending element is employed to satisfy the compatibility of deformation and equilibrium of resultant forces and moments at the delamination crack front. A multipoint constraint algorithm is incorporated which leads to unsymmetric stiffness matrices. The modified Hertzian contact law is employed to simulate the contact force and the time dependent equations are solved by Newmark's time integration algorithm. Parametric studies are conducted with respect to triggering parameters like twist angle, target displacement, initial velocity of striker considering low velocity centrally impacted spherical mass.

Coupling between the in-plane and lateral tape dynamics in high capacity linear tape transport systems

Lateral tape motion (LTM) is defined as the lateral deviation of the tape from its prescribed strain-free state path. The effects of tape, roller, guide or reel imperfections on the LTM are reasonably well understood. Yet, the effects of disturbances in the longitudinal (in-plane) direction, which can couple into LTM, have not been well described. Longitudinal tape vibrations can be due to, but not limited to a) tension dynamics due to servo control of the tape reels, and/or tension impulses due to the unwinding of tape layers that experience sticking. The problem is further complicated due to the uncertainty of the tape length on the downstream side, as a result of the “floating layers” in the take-up reel. In this work, the equations of motion of the longitudinal and lateral tape motion are derived from first principles. The coupling due to non-linear longitudinal strain is considered. The equations of motion are solved by using the finite element method, and an explicit time integration algorithm. The entire tape path is modeled directly, where the interaction of the tape with the recording head and the guides are represented as concentrated forces, and moments. The effects of disturbances, typical for a tape transport system, on the coupling or lack there off are investigated.

Are There Any Differences in Seismic Performance Evaluation Criteria for Concrete Arch Dams

Differences between stress-based and strain-based criteria are investigated in seismic performance evaluation of the arch dams in time domain. A numerical model of the coupled dam-reservoir-foundation system is prepared with the finite element technique. Reservoir is modeled using the Eulerian approach as a compressible domain, and the foundation rock is assumed to be massless. Dynamic equilibrium equations for the coupled system are solved using Newmark’s time integration algorithm. Seismic performance of the arch dam is evaluated according to parameters such as demand-capacity ratio, cumulative inelastic duration and overstressed (or overstrained) areas obtained from linear elastic analyses. The results show, although there are some similarities between stress-based and strain-based criteria, evaluation of the performance based on the strain gives different results which can be led to different decision making in dam safety related projects.

Improving the numerical efficiency for 3D enhanced assumed strain finite elements in explicit time integration

AbstractExplicit time integration has proven to be highly suitable for the simulation of very fast processes and highly nonlinear applications. As is well known the time integration algorithm is only stable if the Courant‐Friedrichs‐Lewy criterion is fulfilled leading usually to very small time step sizes. As a result the efficiency of the time integration algorithm is dictated by the computational time of element operations. In structural mechanics low order finite elements are attractive due to their efficiency but introduce artificial stiffness effects. To resolve the so‐called locking effects a transient description – suitable for the solution in the context of explicit time integration – of the method of enhanced assumed strains (EAS) is formulated utilizing a generalization of the method of incompatible modes (IM). The element routines are computationally realised applying symbolic programming. The functionality and achieved efficiency are demonstrated on a numeric example. (© 2013 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)