Implicit Time Integration Algorithm Research Articles

Magneto-viscoelastic rod model for hard-magnetic soft rods under 3D large deformation: Theory and numerical implementation

The main purpose of this work is to develop a three-dimensional (3D) viscoelastic rod model for hard-magnetic soft (HMS) rods under large deformation which are widely used active structures in soft robotics. To do so, the Simo’s viscoelasticity theory has been rationally incorporated into the geometrically exact 3D curved rod model. The proposed model includes the deformation modes of axial tension, shear, bending, and torsion, which is applicable to the HMS rods with arbitrarily initial curved and twisted geometries under 3D large deformation. The viscoelastic constitutive equations of the HMS rod in the present formulation are formulated, which include the general relaxation functions. To obtain the expression for the magnetic load, the rotation-based magnetic free energy density is introduced, and the governing equations of the HMS rod with magnetic load and body force are presented. To obtain the numerical implementation, an implicit time integration algorithm that simply extends the generalized-α method for the rotation group, and the corresponding variational formulation and its linearization of the rod model are derived. To validate the model, five numerical examples, including 2D dynamic buckling, 3D static, and 3D dynamic problem are presented. The dynamic problems include the dynamic snap-through behavior of a bistable HMS arch and damped oscillation of a quarter arc cantilever under 3D deformation. The simulation results show good agreement with the results reported in the literature.

A re‐evaluation of overshooting in time integration schemes: The neglected effect of physical damping in the starting procedure

AbstractAn important property of implicit time integration algorithms for structural dynamics is their tendency to “overshoot” the exact solution in the first few steps of the computed response due to high‐frequency components in the initial excitations. The typical analysis technique for overshooting involves the study of asymptotic response of the algorithm's first step in the limiting high frequency case. This article finds that the prior analysis of overshooting in much of the engineering literature is incomplete in that it neglects the effect of physical damping. With physical damping included, first‐order overshooting components enter into several well‐known time integration algorithms which were previously thought to exhibit zero‐order overshooting in displacement. The Newmark method, Wilson‐ method, Bazzi‐ method, HHT‐ method, WBZ‐ method, and three parameter optimal/generalized‐ method are analyzed, as well as the generalized single‐step single‐solve (GSSSS) framework which encompasses all of the prior schemes and other new and optimal algorithms and designs based upon the issues under consideration. The additional overshooting component is eliminated in the novel amended GSSSS V0 family (which is noteworthy and cannot be derived by conventional means), while the numerically dissipative schemes in the GSSSS U0 family (encompassing traditional methods such as the HHT‐ method, WBZ‐ method, and three parameter optimal/generalized‐ method among other new algorithm designs) are shown to be irremediable as the additional overshooting component from physical damping enters into the second step of the response, which is a wholly new finding. Numerical verifications of the overshooting analysis are performed for SDOF and MDOF structures with and without physical damping, and practical recommendations are given for the solution of MDOF problems on the basis of algorithm design and selection for various initial conditions.

Development of a 3D fluid-saturated element for dynamic analysis of two-phase media in ABAQUS based on u-U formed equations

The commonly used finite element software does not provide a three-dimensional (3D) fluid-saturated element available for dynamic analysis of two-phase media, which extremely limits their advantages in solving complicated engineering problems. This paper develops a user element for 3D dynamic analysis of fluid-saturated porous media in ABAQUS based on the u-U formed equations. The dynamic governing formulations are firstly revisited, and an implicit time integration algorithm is adopted to solve the dynamic equations. Then, the major development steps, the structure of the subroutine file, and the keywords directly related to the user element in the input file are carefully presented for the application of the user element. Moreover, the equivalent linear approach is introduced with a detailed realization procedure to consider the soil nonlinearity. Finally, the effectiveness and accuracy of the developed user element are validated by comparing the results with those from four different analytical and numerical solutions. The proposed numerical method with user element has a great potential in modeling practical engineering problems due to the prominent advantages of ABAQUS in simulating complicated materials and irregular geometrics.

Constitutive modeling, computational implementation and material parameter identification for polymeric foams considering density and foaming direction

Polyurethane foam (PUF) is a porous material and significantly affected by compressive loads. Under a uniaxial compressive load, the PUF exhibits various types of material nonlinear behavior under varying conditions, such as density, foaming direction, and strain rate. This study investigates the nonlinear behavior of a PUF through phenomenological approach. A Frank–Brockman–Zairi elasto-viscoplastic constitutive model was adopted to investigate the behavior of PUFs with different densities, directions, and strain rates when subjected to compressive loads. The proposed model was discretized using the implicit time integration algorithm and implemented into a user-defined subroutine of ABAQUS. In this study, parameter determination method was also implemented to identify material parameters containing the constitutive equations. Furthermore, the detailed process of determining the material parameters containing the shape of the work-hardening rate–stress curve dependent on the type of material behavior was presented. Consequently, the macroscopic material response of the PUF, such as the stress–strain curve, can be predicted based on the proposed constitutive model. In future studies, the mechanical behavior of porous materials can be analyzed more successfully by supplementing the constitutive model. This can allow the model to simulate the complex plastic flow of porous materials after yielding that is presently difficult to implement.

Implicit Large Eddy simulations of turbulent flow around a square cylinder at [formula omitted

In this paper, the Implicit Large-Eddy Simulation (ILES) is investigated on the flow around a square cylinder incorporating an unstructured Weighted Essential Non-Oscillatory (WENO) reconstruction method for a Reynolds number of 22,000. Simulations are undertaken in the framework of open-source package OpenFOAM and additional implicit 2nd/3rd-order WENO scheme on the convective term of the viscous incompressible Navier-Stokes Equations. A 2nd-order Euler implicit time integration and Pressure-Implicit Splitting-Operator (PISO) algorithm is used to for the pressure-velocity coupling. Conventional LES with Wall Adapting Local Eddy Viscosity (WALE) model is also carried out as a baseline. The results are compared to high fidelity experiment, DNS data and conventional LES with dynamic Smagorinsky model from previous work. Results show favorable performance for ILES with 3rd-order WENO scheme compared with the conventional LES with dynamic Smagorinsky model and similar performance against LES with WALE model. Results also show acceptable predictions over time-averaged statistics with less computational effort for the ILES of 2nd-order WENO scheme. Shear layer flow analysis suggests that both ILES and LES face similar challenges with small quantities, such as shear stress. Finally, simulations are capturing von Kármán vortex, Kelvin-Helmholtz instability and induced frequency changes.

Time Integration Algorithms for Elasto-Viscoplastic Models with Multiple Hardening Laws for Geomaterials: Enhancement and Comparative Study

To describe the behaviours of geomaterials such as time-dependency, anisotropy and destructuration, multiple hardening parameters and laws are generally needed for application in advanced elasto-viscoplastic models. Time integration with stress updating is a key step in the application of elasto-viscoplastic models to engineering practice. However, the robustness of time integration algorithms for such complicated models has rarely been studied, creating difficulties in selecting and improving algorithms. This paper focuses on use of three typical implicit time integration algorithms—Katona, Stolle and cutting plane—for integration of an advanced elasto-viscoplastic model. First, all selected algorithms are improved to fit the characteristics of the advanced model with multiple hardening parameters and are combined with adaptive substepping procedures to enhance their performance. Then a step-changed undrained triaxial test is simulated at the integration point level, on the basis of which variations in iteration number and relative error of stresses with step size are investigated and compared. Furthermore, the advanced model using different algorithms is implemented into finite element code, with global convergence and calculation time investigated and compared for two boundary value problems: a biaxial test and an embankment. All comparisons demonstrate that the modified cutting plane algorithm with substepping is the most robust and efficient one, followed by the modified Stolle with substepping and the modified Katona with substepping, for an advanced model with multiple hardening parameters.

Fixed-Point Fluid structure interaction analysis BASED ON geometrically exact approach

The fluid structure interaction analysis for structures exhibiting large deformations is carried out by using a strong coupling method, in which a fixed point method with Aitken’s dynamic relaxation is employed to accelerate convergence of the coupling iteration, and geometrically exact beam approach initiated by Simo is adopted to simulate the nonlinear flexible beam models. An improved implicit time integration algorithm is given to improve the computation accuracy of structural dynamics. To verify the validity of the fixed-point method in the compressible flows which is usually used in incompressible fluid, it is applied for flutter analysis of AGARD 445.6 wing in the transonic regime. The case of flow-induced vibration of a flexible beam demonstrates that the approach based on geometrically exact beam theory is suitable for the fluid structure interaction analysis and the fixed-point method with Aitken’s relaxation is of great efficiency and robustness in the FSI computation.

Investigation of Density- and Strain Rate-dependent Strain Hardening-softening-coupled material Behavior of Polyurethane Foams using Elasto-viscoplastic Constitutive Model

Polyurethane foam (PUF) is one of the most well-known cellular materials and is widely employed in various industrial and biomedical fields thanks to its many advantages. These include mechanical and material characteristics such as low density and thermal conductivity, and high specific elastic modulus and strength. Despite of these advantages, the PUF has extremely complex material nonlinearity, with changes in density and strain rate, which is a major obstacle to material design and the application of PUF-based structures. PUF has elasto-viscoplastic behavior including three stages of material features, linear elasticity, softening/plateau with stress drop and densification. These phenomena depend strongly on strain rate and density. Therefore, in this study, a phenomenological constitutive model, namely, an elasto-viscoplastic model, was proposed to describe the density- and strain rate-dependent material nonlinear behavior of PUF. The yield surface-independent plastic multiplier, and the hardening- and softening-associated internal state variables proposed by Frank and Brockman, and Zairi et al. were adopted in the constitutive model, respectively. The proposed constitutive model was discretized using the implicit time integration algorithm and was implemented into a user-defined subroutine of the commercial finite element analysis program, ABAQUS. At the same time, a deterministic identification method for material parameters of the constitutive model was introduced to predict the precise material response of PUF under arbitrary densities and strain rates. To do this, the three-dimensional constitutive model was contracted to a one-dimensional equation, and the explicit equation for each material parameter was derived. Then, the strain hardening- and softeningdependent material parameters were calculated using experimental results, such as the work hardening ratestress curve and the yield stress-strain rate curve. After analyzing the obtained material parameters, it was found that the material parameters were strongly dependent on the density and the strain rate. Consequently, the macroscopic material response of PUF, such as a uniaxial compressive stress-strain curve, can be predicted based on the proposed method in this study.

A coupled time integration algorithm for discontinuous deformation analysis using the numerical manifold method

SummaryTo improve the computational efficiency of the numerical manifold method for discontinuous deformation simulations, a spatial‐domain coupled explicit‐implicit time integration algorithm is proposed. A subdomain partition algorithm based on a super manifold element is developed for the numerical manifold method to simulate dynamic motions of blocky rock mass. In different subdomains, explicit or implicit time integration method is employed respectively based on its contact and motion status. These subdomains interact through assembling the corresponding explicit or implicit time integration‐based matrices of different rock blocks. The computational efficiency of the discontinuity system under dynamic loading is improved by partially diagonalizing the global matrices. Two verification examples of a sliding block along an inclined plane under a horizontal acceleration excitation and a multiblock system acted on by dynamic forces are studied to examine the accuracy of the proposed numerical method, respectively. A highly fractured rock mass situated on an inclined slope subjected to seismic excitations is then studied to show the computational efficiency of the developed algorithm. The simulated results are in good agreement with those from the versions using purely implicit or explicit time integration algorithm for the numerical manifold method. The computational efficiency is shown to be higher using the proposed algorithm, which demonstrates its potential for application in dynamic analysis of highly fractured rock masses.

An improved implicit time integration algorithm: The generalized composite time integration algorithm

The weighted residual method is employed to develop one- and two-step time integration schemes. Newly developed time integration schemes are combined to obtain a new second-order accurate implicit time integration algorithm whose computational structure is similar to the Bathe method (Bathe and Noh, 2012). The newly developed algorithm can control algorithmic dissipation in the high frequency limit through the optimized weighting parameters. It contains only one free parameter, and always provides an identical effective stiffness matrix to the first and second sub-steps in linear analyses, which is not provided in the algorithm proposed by Kim and Reddy (2016). Various nonlinear test problems are used to investigate performance of the new algorithm in nonlinear analyses.

A New Family of Higher-Order Time Integration Algorithms for the Analysis of Structural Dynamics

For the development of a new family of implicit higher-order time integration algorithms, mixed formulations that include three time-dependent variables (i.e., the displacement, velocity, and acceleration vectors) are developed. Equal degree Lagrange type interpolation functions in time are used to approximate the dependent variables in the mixed formulations, and the time finite element method and the modified weighted-residual method are applied to the velocity–displacement and velocity–acceleration relations of the mixed formulations. Weight parameters are introduced and optimized to achieve preferable attributes of the time integration algorithms. Specific problems of structural dynamics are used in the numerical examples to discuss some fundamental limitations of the well-known second-order accurate algorithms as well as to demonstrate advantages of using the developed higher-order algorithms.

An efficient finite differences method for the computation of compressible, subsonic, unsteady flows past airfoils and panels

A finite differences scheme is proposed in this work to compute in the time domain the compressible, subsonic, unsteady flow past an aerodynamic airfoil using the linearized potential theory. It improves and extends the original method proposed in this journal by Hariharan, Ping and Scott [1] by considering: (i) a non-uniform mesh, (ii) an implicit time integration algorithm, (iii) a vectorized implementation and (iv) the coupled airfoil dynamics and fluid dynamic loads. First, we have formulated the method for cases in which the airfoil motion is given. The scheme has been tested on well known problems in unsteady aerodynamics —such as the response to a sudden change of the angle of attack and to a harmonic motion of the airfoil— and has been proved to be more accurate and efficient than other finite differences and vortex-lattice methods found in the literature. Secondly, we have coupled our method to the equations governing the airfoil dynamics in order to numerically solve problems where the airfoil motion is unknown a priori as happens, for example, in the cases of the flutter and the divergence of a typical section of a wing or of a flexible panel. Apparently, this is the first self-consistent and easy-to-implement numerical analysis in the time domain of the compressible, linearized coupled dynamics of the (generally flexible) airfoil–fluid system carried out in the literature. The results for the particular case of a rigid airfoil show excellent agreement with those reported by other authors, whereas those obtained for the case of a cantilevered flexible airfoil in compressible flow seem to be original or, at least, not well-known.

Coupled Time Domain Integration Algorithm for Numerical Manifold Method

AbstractA temporal coupled explicit–implicit time integration algorithm is proposed to improve the computational efficiency of the numerical manifold method (NMM) for seismic stability analysis of rock slope. It includes a coupled time integration scheme, a phase transfer criterion, and an associated contact algorithm. To calibrate the proposed algorithm, a block sliding along a slope under seismic excitation and a block rocking under half-sine pulse shaking are simulated. The traditional limit equilibrium method (LEM) for slope stability analysis determines the factor of safety (FoS) without considering the time-dependent effect. The developed algorithm is able to simulate jointed rock slope seismic stability taking advantage of the NMM for continuous and discontinuous deformation analysis. An open-pit mine slope excited by the El Centro earthquake wave is studied. The simulated results are in good agreement with the results based on the traditional NMM, whereas the computational efficiency is improved s...

Construction of minimum energy high-order Helmholtz bases for structured elements

We present a construction procedure for high-order expansion bases for structured finite elements specific for the operator under consideration. The procedure aims to obtain bases in such way that the condition numbers for the element matrices are almost constant or have a moderate increase in terms of the polynomial order. The internal modes of the mass and stiffness matrices are made simultaneously diagonal and the minimum energy concept is used to make the boundary modes orthogonal to the internal modes. The performance of the proposed bases is compared to the standard basis using Jacobi polynomials. This is performed through numerical examples for Helmholtz problem and transient linear elasticity employing explicit and implicit time integration algorithms and the conjugate gradient method with diagonal, SSOR and Gauss-Seidel pre-conditioners. The sparsity patterns, conditioning and solution costs are investigated. A significant speedup and reduction in the number of iterations are obtained when compared to the standard basis.

A geometrically exact formulation for three-dimensional numerical simulation of the umbilical cable in a deep-sea ROV system

This paper proposes a geometrically exact formulation for three-dimensional static and dynamic analyses of the umbilical cable in a deep-sea remotely operated vehicle (ROV) system. The presented formulation takes account of the geometric nonlinearities of large displacement, effects of axial load and bending stiffness for modeling of slack cables. The resulting nonlinear second-order governing equations are discretized spatially by the finite element method and solved temporally by the generalized-α implicit time integration algorithm, which is adapted to the case of varying coefficient matrices. The ability to consider three-dimensional union action of ocean current and ship heave motion upon the umbilical cable is the key feature of this analysis. The presented formulation is firstly validated, and then three numerical examples for the umbilical cable in a deep-sea ROV system are demonstrated and discussed, including the steady configurations only under the action of depth-dependent ocean current, the dynamic responses in the case of the only ship heave motion, and in the case of the combined action of the ship heave motion and ocean current.

An unconditionally stable implicit time integration algorithm: Modified quartic B-spline method

In Ref. [1] an effective conditionally stable explicit time integration scheme using quartic B-spline function was proposed for solving the problems in structural dynamics. The current paper presents a scheme where this method is developed to an implicit unconditionally stable time integration method. Using quartic B-spline basis function, this method gained second order of acceleration at each time-step. In this research, in order to applying the stabilization process, first, a series of implicit standard formulas were derived from previous formulation. Then after inserting two controlling parameters γ and β in the standard formulas, unconditional stability is guaranteed.

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.

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.

A high-order time-domain transmitting boundary for cylindrical wave propagation problems in unbounded saturated poroelastic media

Based on the u–p formulation of Biot equation with an assumption of zero permeability coefficient, a high-order transmitting boundary is derived for cylindrical elastic wave propagation in infinite saturated porous media. By this transmitting boundary the total stresses on the truncated boundaries of a numerical model, such as a finite element model, are replaced by a set of spring, dashpot and mass elements, with some additionally introduced auxiliary degrees of freedom. The transmitting boundaries are incorporated into the DIANA SWANDYNE II program and an unconditionally stable implicit time integration algorithm is adopted. Despite the assumption made in the derivation of the transmitting boundary, numerical examples show that it can provide highly accurate results for cylindrical elastic wave propagation problems in infinite saturated porous medium in case the u–p formulation is applicable. Although the direct applications of the proposed transmitting boundary to general two dimensional wave problems in infinite saturated porous media are not highly accurate, acceptable accuracy can still be achieved by placing the transmitting boundary at relatively large distance from the wave source.

Dynamic cohesive crack propagation modelling using the scaled boundary finite element method

ABSTRACTThis study develops a scaled boundary finite element method (SBFEM)‐based approach for modelling fast cohesive crack propagation in quasi‐brittle materials subjected to transient dynamic loadings. In this approach, the elastic bulk material is modelled by SBFEM subdomains and the cracks by nonlinear cohesive interface finite elements that are automatically inserted by a remeshing procedure. The global equation system is solved using an implicit time integration algorithm. Because all the solutions (displacements, stresses, velocities, accelerations) are semi‐analytical in an SBFEM subdomain, this approach offers a few advantages over other methods, such as accurate calculation of dynamic stress intensity factors and T‐stress without using fine crack‐tip meshes, simpler remeshing, more accurate and efficient mesh mapping, and the need of much fewer degrees of freedom for the same accuracy. The approach is validated by modelling two concrete beams under impact, subjected to mode‐I and mixed‐mode fracture, respectively.