Time Integration Procedure Research Articles

Energy-consistent integration of multibody systems with friction

The main goal of the present contribution is to establish an energy consistent time integration procedure for multibody systems with friction. Based upon a rotationless formulation for multibody systems, we use the coordinate augmentation technique in order to implement frictional phenomena such as joint friction. Within this contribution we use the viscous friction assumption as a classical friction model incorporated in the joints. We start with a classical ad hoc model for damping and extend the formulation to multibody systems, dealing finally with an underactuated control problem of a three bar linkage with applied joint friction.

A large deformation theory for rate-dependent elastic–plastic materials with combined isotropic and kinematic hardening

We have developed a large deformation viscoplasticity theory with combined isotropic and kinematic hardening based on the dual decompositions F = F e F p [Kröner, E., 1960. Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Archive for Rational Mechanics and Analysis 4, 273–334] and F p = F en p F dis p [Lion, A., 2000. Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. International Journal of Plasticity 16, 469–494]. The elastic distortion F e contributes to a standard elastic free-energy ψ ( e ) , while F en p , the energetic part of F p , contributes to a defect energy ψ ( p ) – these two additive contributions to the total free energy in turn lead to the standard Cauchy stress and a back-stress. Since F e = FF p - 1 and F en p = F p F dis p - 1 , the evolution of the Cauchy stress and the back-stress in a deformation-driven problem is governed by evolution equations for F p and F dis p – the two flow rules of the theory. We have also developed a simple, stable, semi-implicit time-integration procedure for the constitutive theory for implementation in displacement-based finite element programs. The procedure that we develop is “simple” in the sense that it only involves the solution of one non-linear equation, rather than a system of non-linear equations. We show that our time-integration procedure is stable for relatively large time steps, is first-order accurate, and is objective.

Inclusion of Transverse Shear Deformation in a Beam Element Based on the Absolute Nodal Coordinate Formulation

In this study, a procedure to account for transverse shear deformation in the absolute nodal coordinate formulation is presented. In the absolute nodal coordinate formulation, shear deformation is usually defined by employing the slope vectors in the element transverse direction. This leads to the description of deformation modes that are, in practical problems, associated with high frequencies. These high frequencies, in turn, complicate the time integration procedure burdening numerical performance. In this study, the description of transverse shear deformation is accounted for in a two-dimensional beam element based on the absolute nodal coordinate formulation without the use of transverse slope vectors. In the introduced shear deformable beam element, slope vectors are replaced by vectors that describe the orientation of the beam cross-section. This procedure represents a simple enhancement that does not decrease the accuracy or numerical performance of elements based on the absolute nodal coordinate formulation. Numerical results are presented in order to demonstrate the accuracy of the introduced element in static and dynamic cases. The numerical results obtained using the introduced element agree with the results obtained using previously proposed shear deformable beam elements.

Geometric properties of projective constraint violation stabilization method for generally constrained multibody systems on manifolds

During numerical forward dynamics of constrained multibody systems, a numerical violation of system kinematical constraints is the important issue that has to be properly treated. In this paper, the stabilized time-integration procedure, whose constraint stabilization step is based on the projection of integration results to underlying constraint manifold via post-integration correction of the selected coordinates is discussed. A selection of the coordinates is based on the optimization algorithm for coordinates partitioning. After discussing geometric background of the optimization algorithm, new formulae for optimized partitioning of the generalized coordinates are derived. Beside in the framework of the proposed stabilization algorithm, the new formulae can be used for other integration applications where coordinates partitioning is needed. Holonomic and non-holonomic systems are analyzed and optimal partitioning at the position and velocity level are considered further. By comparing the proposed stabilization method to other projective algorithms reported in the literature, the geometric and stabilization issues of the method are addressed. A numerical example that illustrates application of the method to constraint violation stabilization of non-holonomic multibody system is reported.

Mixed finite element solution of time-dependent problems

A mixed formulation of the finite element method is used to establish a higher-order incremental method for the solution of second-order/hyperbolic problems. The displacement, the velocity and, optionally, the acceleration fields are approximated independently in time using hierarchical bases. The time approximation criterion preserves hyperbolicity in the sense that it replaces the solution of hyperbolic problems by the solution of uncoupled Helmholtz-type elliptic problems, which can be subsequently solved using the alternative methods currently in use for discretization of the space dimension. The development of the time integration procedure, the characterization of its performance in terms of stability, accuracy and convergence are illustrated using a polynomial time basis. In order to stress the fact that the procedure can be implemented using alternative time bases, a wavelet system is used in the solution of nonlinear, parabolic and hyperbolic problems. The method is well-suited to parallel processing and to large time stepping. The extension of its application to the solution of other than linear second-order/hyperbolic problems is discussed.

Performance of Composite Implicit Time Integration Scheme for Nonlinear Dynamic Analysis

This paper presents a simple implicit time integration scheme for transient response solution of structures under large deformations and long‐time durations. The authors focus on a practical method using implicit time integration scheme applied to structural dynamic analyses in which the widely used Newmark time integration procedure is unstable, and not energy‐momentum conserving. In this integration scheme, the time step is divided in two substeps. For too large time steps, the method is stable but shows excessive numerical dissipation. The influence of different substep sizes on the numerical dissipation of the method is studied throughout three practical examples. The method shows good performance and may be considered good for nonlinear transient response of structures.

A finite element model for linear viscoelastic behaviour of pultruded thin-walled beams under general loadings

A finite element model for orthotropic thin-walled beams subject to long-term loadings is presented. The hypothesis, rather usual for thin-walled beams, of cross-sections remaining undistorted in their own planes after deformation is introduced, so reducing the number of d.o.f.’s and, consequently, the computational effort of the analysis. The model is used to perform linear viscoelastic analysis of prismatic beams with general cross-sections, i.e., open, closed or multi-cell. As far as the constitutive viscoelastic law is concerned, a generalized linear Maxwell model is adopted. Making use of the exponential algorithm, differential equations are written in incremental form and integration is performed adopting time intervals of variable length. Numerical examples are finally presented, concerning glass-fibre pultruded shapes under long-term loadings. Displacement evolution with time and stress redistribution adopting different creep laws are presented. Convergence features of the proposed finite element and time integration procedure are also shown.

Designing a positive second-order implicit time integration procedure for unsteady turbulent flows

A positivity preserving implicit procedure for first order time integration of two-equation turbulence models is extended to second-order time integration utilizing dual-time stepping. Because the time derivative of the turbulence quantities may behave as a negative source term it is treated implicitly. The procedure is applied to a non-linear k – ϵ turbulence model using an unstructured flow solver. Numerical simulations of the unsteady flow about a square cylinder are conducted to demonstrate the positivity characteristics of the turbulence quantities and the robustness of the procedure and of the flow solver. Results from the numerical simulations favorably compare with results from experiments.

A Survey in Mathematics for Industry An efficient method for the numerical simulation of magneto-mechanical sensors and actuators

The dynamic behaviour of magneto-mechanical sensors and actuators can be completely described by Maxwell's and Navier-Lamé's partial differential equations (PDEs) with appropriate coupling terms reflecting the interactions of these fields and with the corresponding initial, boundary and interface conditions. Neglecting the displacement currents, which can be done for the classes of problems considered in this paper, and introducing the vector potential for the magnetic field, we arrive at a system of degenerate parabolic PDEs for the vector potential coupled with the hyperbolic PDEs for the displacements.Usually the computational domain, the finite element discretization, the time integration, and the solver are different for the magnetic and mechanical parts. For instance, the vector potential is approximated by edge elements whereas the finite element discretization of the displacements is based on nodal elements on different meshes. The most time consuming modules in the solution procedure are the solvers for both, the magnetical and the mechanical finite element equations arising at each step of the time integration procedure. We use geometrical multigrid solvers which are different for both parts. These multigrid solvers enable us to solve quite efficiently not only academic test problems, but also transient 3D technical magneto-mechanical systems of high complexity such as solenoid valves and electro-magnetic-acoustic transducers. The results of the computer simulation are in very good agreement with the experimental data.

Parallel computing for seismic response analysis of immersed tunnel with domain decomposition

PurposeTaking into account the long‐term influences of the non‐linear behavior of the material as well as the large deformation and contact conditions, the limiting factors of the computer simulation are the computer runtime and the memory requirement during solution of seismic response analysis for immersed tunnel. This research aims to overcome these problems.Design/methodology/approachThis research deals with parallel explicit finite element simulation with domain decomposition for seismic response analysis of immersed tunnel, which is the non‐linear and time‐dependent behavior of complex structures in engineering. A domain decomposition method based on parallel contact algorithm and dynamic‐explicit time integration procedure are used, and the latter is used for the solution of the semi‐discrete equations of motion, which is very suited for parallel processing. Using the high performance computer SGI Onyx3800, the seismic response analysis of the immersed tunnel in Shanghai is processed with more than 1.2 million nodes and more than 1 million elements in final finite element model.FindingsThe results show numerical scalability of this algorithm and reveal the dangerous joints in this immersed tunnel under Tangshan seismic acceleration, and it could also provide references for the antiseismic design of the immersed tunnel.Originality/valueWith the increasing demands in the scale, accuracy and speed of numerical simulation in geotechnical engineering, parallel computing has its great application in this area. This paper fulfils an identified method need, and it is believed more and more research work will be devoted to this research field in the near future.

High-order time integration applied to metal powder plasticity

The present article treats two objectives. In the first investigation attention is focused on the application of time-adaptive finite elements formulated on the basis of a high-order time integration procedure on a constitutive model for compressible finite strain viscoplasticity for metal powder. In this connection, it has to be emphasized that the integration procedure is not only applied to the evolution equations on Gauss-point level but on the total system of differential–algebraic equations resulting from the application of the vertical line method on the quasi-static finite element equations. The specific application emerges from the field of metal powder compaction. Particular studies are carried out using stiffly accurate, diagonally implicit Runge–Kutta methods in combination with the Multilevel-Newton algorithm for solving the DAE-system. In this respect, the effort vs. accuracy behavior is investigated which is also related to order reduction known in elastoplasticity. The second topic treats the local stress algorithm for taking into account the yield function based finite strain viscoplasticity model, where the classical Newton–Raphson method fails. This is the reason why most constitutive models of powder materials are implemented into explicit finite element codes. Thus, the proposed investigations compare different methods in view of a stable and efficient integration process in implicit finite element formulations.

An implicit high order finite volume scheme for the solution of 3D Navier–Stokes equations with new discretization of diffusive terms

The computation of three-dimensional viscous flows on complex geometries requiring distorted meshes is of great interest. This paper presents a finite volume solver using a quadratic reconstruction of the unknowns for the advective fluxes computation, and a conservative and consistent discretization of the diffusive terms, based on an extended version of the Coirier's diamond path. A fully implicit time integration procedure is employed with a preconditioned matrix-free GMRES solver.

A note on time integrators in water-wave simulations

The importance of a suitable temporal integrator for fully nonlinear simulations of surface gravity waves is emphasized. Via numerical examples, it is demonstrated that constant-step procedures are inefficient. This relates to the practice of energy-conserving symplectic integration, assuming constant time steps, and is compared to direct numerical simulations using Runge–Kutta integrators with variable time-step control. It is concluded that the latter with automatic variable time-step control is the more efficient, and should be applied. The practice and efficiency of a stabilization procedure for the time-step selection is described and illustrated. An important point of the method is that the linear part of the prognostic equations is integrated analytically, which means that this part is obtained to machine presicion for any (large) time step (time interval). The evolution and instabilities of highly nonlinear water waves in three dimensions are exemplified through an accurate and efficient time-integration procedure.

Conserving energy and momentum in nonlinear dynamics: A simple implicit time integration scheme

We focus on a simple implicit time integration scheme for the transient response solution of structures when large deformations and long time durations are considered. Our aim is to have a practical method of implicit time integration for analyses in which the widely used Newmark time integration procedure is not conserving energy and momentum, and is unstable. The method of time integration discussed in this paper is performing well and is a good candidate for practical analyses.

Convergence Acceleration of an Upwind Least Squares Finite Difference based Meshless Solver

One of the important trends in the present day CFD research is the development of new class of flow solvers called meshless solvers. All that this class of solvers require is a distribution of points in the computational domain. The development of upwind least squares finite difference method can be considered as a landmark event in the advances being made in the area of meshless solvers. In this paper, we propose an implicit time integration methodology for upwind least squares finite difference procedure. The idea of matrix-free implicit procedure in the framework of finite volume solver has been exploited in the present work to obtain a cheap and robust implicit time integration procedure. The present formulation shows excellent convergence acceleration over the explicit upwind least squares finite difference procedure.

An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes - I. The two-dimensional isotropic case with external source terms

SUMMARY We present a new numerical approach to solve the elastic wave equation in heterogeneous media in the presence of externally given source terms with arbitrary high-order accuracy in space and time on unstructured triangular meshes. We combine a discontinuous Galerkin (DG) method with the ideas of the ADER time integration approach using Arbitrary high-order DERivatives. The time integration is performed via the so-called Cauchy-Kovalewski procedure using repeatedly the governing partial differential equation itself. In contrast to classical finite element methods we allow for discontinuities of the piecewise polynomial approximation of the solution at element interfaces. This way, we can use the well-established theory of fluxes across element interfaces based on the solution of Riemann problems as developed in the finite volume framework. In particular, we replace time derivatives in the Taylor expansion of the time integration procedure by space derivatives to obtain a numerical scheme of the same high order in space and time using only one single explicit step to evolve the solution from one time level to another. The method is specially suited for linear hyperbolic systems such as the heterogeneous elastic wave equations and allows an efficient implementation. We consider continuous sources in space and time and point sources characterized by a Delta distribution in space and some continuous source time function. Hereby, the method is able to deal with point sources at any position in the computational domain that does not necessarily need to coincide with a mesh point. Interpolation is automatically performed by evaluation of test functions at the source locations. The convergence analysis demonstrates that very high accuracy is retained even on strongly irregular meshes and by increasing the order of the ADER‐DG schemes computational time and storage space can be reduced remarkably. Applications of the proposed method to Lamb’s Problem, a problem of strong material heterogeneities and to an example of global seismic wave propagation finally confirm its accuracy, robustness and high flexibility.

Finite element modeling of crystal plasticity with grains shaped as truncated octahedrons

Different modeling strategies are tested for the prediction of texture development and microscopic strain heterogeneity in cold-rolled ULC steel and in multiphase steel under uniaxial tension. The polycrystalline aggregate is represented by a finite element mesh that is loaded under periodic boundary conditions. Grains are shaped as cubes or as truncated octahedrons, defining three levels of mesh refinement. Simulations rely on a simplified implementation of crystal plasticity, in which elastic strains are considered infinitesimal. The constitutive law is integrated fully implicitly in a reference frame bounded to the crystal lattice. Accuracy of the time-integration procedure is assessed by referring to some predictions of the crystal plasticity algorithm developed by Kalidindi et al. [Crystallographic texture evolution in bulk deformation processing of FCC metals. J. Mech. Phys. Solid 40 (1992) 537–569]. Then, results of the micro–macro modeling are compared to experimental data. It is found that the simulations with truncated octahedral grains yield improved predictions compared to those with cuboidal grains.

Parallel numerical simulation with domain decomposition for seismic response analysis of shield tunnel

The research presented in this paper deals with parallel explicit finite element simulation with domain decomposition for seismic response analysis of shield tunnel, which is the non-linear and time dependent behaviour of complex structures in engineering. Such simulations have to provide high accuracy in the prediction of deformations and stability, by taking into account the long-term influences of the non-linear behaviour of the material as well as the large deformation and contact conditions. The limiting factors of the computer simulation are the computer run time and the memory requirement during solution of large-scale problems. To overcome these problems, a domain decomposition method and dynamic-explicit time integration procedure are used, and the latter is used for the solution of the semi-discrete equations of motion, which is very suited for parallel processing. Using the high performance computer ShenWei-I, the seismic response of shield tunnel in Shanghai is processed, and the number of nodes and elements in final finite element model exceeded 4.0 million and 3.8 million, respectively. The results reveal the weak position in the shield tunnel under Shanghai seismic wave. The paper provides references for the antiseismic design of the shield tunnel.

Multirate hierarchical time integration for electronic circuits

AbstractElectrical circuits are hierarchically defined, in which electrical components are combined with submodels. These last ones may contain submodels themselves. This construction gives options for dedicated time integration procedures. We describe an extension that allows for a dynamically partitioned multirate time integration that retains a number of hierarchical efficiencies. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Numerical analysis of large deformation processes at elevated temperatures

The authors describe a theoretical model and kinematic approach suitable for the numerical analysis of problems at large deformation finite strains and rotations. The presented formulations are implemented incrementally with a non-linear and rate-dependent constitutive model, adequate to the simulation of high temperature processes. The large deformation kinematic model is based on the additive decomposition of the transformation gradient and the strain rate tensor in thermoelastic and viscoplastic parts. Stress–strain relations leading to the constitutive equations governing finite deformations are formulated on an unrotated frame of reference. These relations are obtained either by polar decomposition of the transformation gradient or by the rotation increment method. Both approaches and algorithms are thoroughly compared and discussed. The flow rule is a function of the equivalent stress and the deviatoric stress tensor, of the temperature field and of a set of internal state variables. The increments of the state variables are calculated with a forward gradient time integration procedure, based on a numerical estimation of the integral of the strain rate tensor. The model was implemented in a three-dimensional finite element code and tested with a set of tensile and shear tests on aluminium alloy specimens. In order to evaluate the performance of the numerical algorithms with elevated temperature processes, tests were done at temperatures ranging from room temperature to 580K. Numerical simulation results are compared with experimental results and shown to be in good agreement. Real material behaviour is fairly well predicted and matches the experimental results obtained with different loading conditions at different temperatures.