Explicit Integration Procedure Research Articles

Explicit and Implicit Integration of Constitutive Equations of Chaboche Isotropic-Kinematic Hardening Material Model

The proper selection of the material model and its parameters in numerical simulations of the material response under loading is a very important task. In the cyclic load one should assume different phenomena, e.g. ratchetting effect, the mean stress relaxation, creep, etc. The Chaboche kinematic hardening model is mainly applied in order to capture the Bauschinger effect due to its good description of the material behaviour under the cyclic loading. The modified Chaboche isotropic-kinematic hardening model is considered in this paper in order to simulate the strain-controlled and stress-controlled uniaxial cyclic tension-compression test. The implicit and explicit integration procedures are tested here, showing advantages and disadvantages of both methods. It is shown that both approaches provide similar results. The implicit integration method of constitutive equations is unconditionally stable and thus less calculation steps are required in comparison to the explicit procedure. The implicit and explicit integration of the Chaboche model including isotropic and kinematic hardening are then implemented for the 3D case in commercial ABAQUS program in the form of the user material procedure. The correctness of results obtained for the user material model is verified by comparison to results for commercially implemented Chaboche model.

Simulation of non-linear structural elastodynamic and impact problems using minimum energy high-order bases

We present the application of simultaneous diagonalization and minimum energy (SDME) high-order finite element modal bases for simulation of transient non-linear elastodynamic problem, including impact cases with Hookean and neo-Hookean hyperelastic materials. The bases are constructed using procedures for simultaneous diagonalization of the internal modes and Schur complement of the boundary modes from the standard nodal and modal bases, constructed using Lagrange and Jacobi polynomials, respectively. The implementation of these bases in a high-order finite element code is straightforward, since the procedure is applied only to the one-dimensional expansion bases. Non-linear transient structural problems with large deformation , hyperelastic materials and impact are solved using the obtained bases with explicit and implicit time integration procedures. Iterative solutions based on preconditioned conjugate gradient methods are considered. The performance of the proposed bases in terms of the number of iterations of pre-conditioned conjugate gradient methods and computational time are compared with the standard nodal and modal bases. The SDME bases are accurate and provide better conditioned equations for iterative solutions in general, leading to substantial reduction in processing time over an order of magnitude compared to other modal bases. Our numerical tests obtained speedups up to 41 using the considered bases when compared to the standard modal basis. Results for a two-dimensional impact problem approximated with the SDME bases showed that the implicit Newmark method performed much better in terms of processing time and contact stress error when compared to the standard explicit central difference method with lumped mass matrix . • Application of simultaneous diagonalization procedure and minimum energy highorder bases to non-linear structural and impact transient problems. • Simple modifications of high-order bases improve conditioning of system of equations obtained from the high-order approximation of the considered problems, with remarkable gains in performance of iterative solvers. • Light variation on the number of pre-conditioned gradient iterations in terms of the polynomial order. • Speed-ups over 40 when compared to the standard modal Jacobi high-order bases used in high-order interpolation.

Comparative Numerical Studies on the Structural Behavior of Buried Pipes Subjected to Extreme Environmental Actions.

Globally, there are several critical infrastructure networks (water and gas networks) whose disruption or destruction would significantly affect the maintenance of vital societal functions, such as the health, safety, security, and social or economic well-being of people. They would also have significant local, regional, and national impacts as a result of the inability to maintain those functions, and would have similar cross-border effects. The main objective of this article is to investigate by comparative numerical studies the structural response of three types of buried pipes made of different materials, primarily steel, concrete, and high-density polyethylene, resulting from the impact of the environment through exceptional external actions, such as explosions at the surface of the land in the vicinity of the laying areas. The dynamic transient analysis of the equation of motion with the application of the explicit integration procedure was performed with the ANSYS numerical simulation program. This study allows designers to solve complex problems related to the quality of the laying ground of water networks to canals. The knowledge accumulated gives us the possibility to correctly specify the optimal economic and technical value of the ratio between the laying depth of pipes and their diameter, the importance of the radius ratio of the pipe and the thickness of its wall, and, importantly, the improvement of the quality of the foundation ground. Following the results obtained, it is estimated that the optimal economic and technical value of the ratio between the laying depth of the pipes (H) and their diameter (D) is 3, regardless of the material from which the pipe is made.

A New Procedure for Nonlinear Dynamic Analysis of Structures Under Seismic Loading Based on Equivalent Nodal Secant Stiffness

This paper presents a dynamic analysis procedure for predicting the responses of large, highly nonlinear, discontinuous structural systems subjected to seismic loading. The concept of equivalent nodal secant stiffness is adopted to diagonalize the conventional stiffness matrix of the structure. With the lumped-mass idealization, the decoupled equilibrium equations of the structure are then solved by the implicit Newmark integration method. Additionally, an incremental-iterative procedure is performed to ensure that the equilibrium conditions are satisfied at the end of each time step. The proposed analysis procedure has the advantages of both the conventional explicit and implicit integration procedures, but with their disadvantages removed. Through extensive applications, the results demonstrate that the proposed procedure is simple and robust for analyzing practical structural systems in terms of computational efficiency and stability.

A new partitioned staggered scheme for flexible multibody interactions with strong inertial effects

In generic flexible multibody interaction problems, the differences in the mass, damping and stiffness of the bodies can be very large. Of particular interest is a small change in the higher mass structure which triggers relatively large inertial effects on the lower mass structure. In this work, a new partitioned staggered time integration scheme is developed and its applicability is extended to the problems involving low mass ratios in coupled multibody problems. The equation of motion of body with larger inertia is integrated by implicit Newmark method, whereas the smaller inertia is solved using a robust self-starting explicit integration procedure. The coupled formulation includes explicit correction terms that adjusts the amount of interfacial velocity, which plays a key role in the stability and accuracy of the simulations. A wide range of mass (10−6–10−1), damping and stiffness (10−3–103) ratios are chosen to assess the stability and accuracy characteristics of the proposed scheme. The closed-form expressions for the coupling parameter have been constructed for both matching and non-matching time stepping through the Godunov–Ryabenkii normal mode analysis. The stability of the proposed method is observed as a function of the relative properties and temporal discretisation of the coupled system. The time step of the heavier body significantly influences the stability more than the lighter body. To maintain the staggering error minimal, an optimal range of the coupling parameter is identified. The error computed with uniform variation in the time step revealed that the present method is more accurate than the existing methods. The partitioned method is an energy preserving integration method for the dynamic analysis of multibody systems. As a potential application of this scheme, flexible multibody problems in the field of offshore engineering, viz., wave energy converter and floater–mooring systems are presented and validated with experimental measurements.

Three-dimensional finite element simulation and application of high-strength bolts

High-strength structural bolts have been utilized for beam-to-column connections in steel-framed structural buildings. Failure of these components may be caused by the bolt shank fracture or threads stripping-off, documented in the literature. Furthermore, these structural bolts are galvanized for corrosion resistance or quenched-and-tempered in the manufacturing process. This paper adopted the finite element simulation to demonstrate discrete mechanical performance for these bolts under tensile loading conditions, the coated and uncoated numerical model has been built up for two numerical integration methods: explicit and implicit. Experimental testing and numerical methods can fully approach the failure mechanism of these bolts and their ultimate load capacities. Comparison has also been conducted for two numerical integration methods, demonstrating that the explicit integration procedure is also suitable for solving quasi-static problems. Furthermore, by using precise bolt models in T-Stub, more accurately simulate the mechanical behavior of T-Stub, which will lay the foundation of the mechanical properties of steel bolted joints.

Blast loading influence on load carrying capacity of I-column

The paper presents the multistage numerical analyses procedure for a load carrying capacity assessment of a blast loaded I-column. The procedure determines the ultimate axial load in four stages: static preload, dynamic blast loading, dynamic response after blast loading and a static load carrying capacity assessment of a deformed structure. Preload and assessment of the ultimate axial load were performed using static Newton–Raphson algorithm. Consequently, two dynamic cases were analysed using a commercial code with an explicit integration procedure (central difference method), where the blast load was simulated using Lagrangian–Eulerian domain coupling. The developed method proved to be a very efficient way to assess the load carrying capacity of a column with imperfections caused by dynamic, blast loading. The results of analyses indicated that the column (HKS-300 cross-section) is very resistant to blast loading. High deformation of its structure (close to ultimate strains) caused only 10% reduction of the load carrying capacity. Separation between a web and flanges of the I-column was the only factor that caused the significant loss of carrying capacity. Moreover, not only the mass of a charge influences the structural behaviour but also the explosive shape as well as the initiation point of detonation can significantly change the level of structure damage. Initiation point of detonation at the back of HE heavily increased blast pressure during interaction with the structure and a cylindrical shape of the explosive also increased the energy absorbed by the column. On the other hand, the use of the spherical charge or frontal detonation significantly reduced the destructive capabilities of the explosives.

Evaluation of numerical stress-point algorithms on elastic–plastic models for unsaturated soils with hardening dependent on the degree of saturation

Constitutive models of unsaturated soils, and in particular those based on constitutive variables which include both degree of saturation and suction, are characterised by strong non linearities due to hydromechanical coupling. In this paper, a refined Runge–Kutta–Dormand–Prince explicit algorithm and a fully implicit Euler scheme are compared for the integration of the latter class of models. The explicit and implicit procedures have been tested along different hydromechanical paths, involving various hydraulic and mechanical external control conditions. Accuracy and efficiency of the algorithms have been investigated. The results confirm that substepping is mandatory for the explicit algorithm to converge regardless the initial step size and to remain sufficiently accurate. The value of the incremental hydromechanical work per unit volume was calculated during the explicit integration procedure. The numerical results show that the maximum size of the substep which can be adopted to meet a given tolerance depends on the gradient of the incremental work per unit volume. Therefore, the latter appears a good candidate to identify problematic integration steps in terms of convergence. Accuracy of the implicit algorithm also depends on the chosen step size, although the algorithm proved to be convergent in all the paths analysed.

Rubber Structural Coupon Behaviour Study under Pressure Impulse

This study focuses on the rubber material behaviour assessment under dynamic loading using numerical methods. Consequently, dynamic simulations of the rubber structural coupon subjected to dynamic velocity loading were performed using the explicit integration procedure with central difference scheme with modified time integration of the equation of motion implementation. During investigations two impulse velocities were used and compared for two different constitutive materials: Mooney-Rivlin without rate-dependency and Mat 181 Simplified Rubber which includes strain rate effects. From the obtained results it was noticed that material behaviour in both cases is different and along with different values of velocity the strain rate sensitivity changes.

Full three-dimensional investigation of structural contact interactions in turbomachines

Minimizing the operating clearance between rotating bladed-disks and stationary surrounding casings is a primary concern in the design of modern turbomachines since it may advantageously affect their energy efficiency. This technical choice possibly leads to interactions between elastic structural components through direct unilateral contact and dry friction, events which are now accepted as normal operating conditions. Subsequent nonlinear dynamical behaviors of such systems are commonly investigated with simplified academic models mainly due to theoretical difficulties and numerical challenges involved in non-smooth large-scale realistic models. In this context, the present paper introduces an adaptation of a full three-dimensional contact strategy for the prediction of potentially damaging motions that would imply highly demanding computational efforts for the targeted aerospace application in an industrial context. It combines a smoothing procedure including bicubic B-spline patches together with a Lagrange multiplier based contact strategy within an explicit time-marching integration procedure preferred for its versatility.The proposed algorithm is first compared on a benchmark configuration against the more elaborated bi-potential formulation and the commercial software Ansys. The consistency of the provided results and the low energy fluctuations of the introduced approach underlines its reliable numerical properties. A case study featuring blade-tip/casing contact on industrial finite element models is then proposed: it incorporates component mode synthesis and the developed three-dimensional contact algorithm for investigating structural interactions occurring within a turbomachine compressor stage. Both time results and frequency-domain analysis emphasize the practical use of such a numerical tool: detection of severe operating conditions and critical rotational velocities, time-dependent maps of stresses acting within the structures, parameter studies and blade design tests.

Numerical aspects of a real‐time sub‐structuring technique in structural dynamics

AbstractA time domain coupling technique, involving combined computational and experimental modelling, for vibration analysis of structures built‐up of linear/non‐linear substructures is developed. The study permits, in principle, one or more of the substructures to be modelled experimentally with measurements being made only on the interfacial degrees of freedom. The numerical and experimental substructures are allowed to communicate in real time within the present framework. The proposed strategy involves a two‐stage scheme: the first is iterative in nature and is implemented at the initial stages of the solution in a non‐real‐time format; the second is non‐iterative, employs an extrapolation scheme and proceeds in real time. Issues on time delays during communications between different substructures are discussed. An explicit integration procedure is shown to lead to solutions with high accuracy while retaining path sensitivity to initial conditions. The stability of the integration scheme is also discussed and a method for numerically dissipating the temporal growth of high‐frequency errors is presented. For systems with non‐linear substructures, the integration procedure is based on a multi‐step transversal linearization method; and, to account for time delays, we employ a multi‐step extrapolation scheme based on the reproducing kernel particle method. Numerical illustrations on a few low‐dimensional vibrating structures are presented and these examples are fashioned after problems of seismic qualification testing of engineering structures using real‐time substructure testing techniques. Copyright © 2007 John Wiley & Sons, Ltd.

A novel non‐linearly explicit second‐order accurate L‐stable methodology for finite deformation: hypoelastic/hypoelasto‐plastic structural dynamics problems

AbstractA novel non‐linearly explicit second‐order accurate L‐stable computational methodology for integrating the non‐linear equations of motion without non‐linear iterations during each time step, and the underlying implementation procedure is described. Emphasis is placed on illustrative non‐linear structural dynamics problems employing both total/updated Lagrangian formulations to handle finite deformation hypoelasticity/hypoelasto‐plasticity models in conjunction with a new explicit exact integration procedure for a particular rate form constitutive equation. Illustrative numerical examples are shown to demonstrate the robustness of the overall developments for non‐linear structural dynamics applications. Copyright © 2004 John Wiley & Sons, Ltd.

Numerical analysis and elastic–plastic deformation behavior of anisotropically damaged solids

Abstract The present paper is concerned with the numerical modelling of the large elastic–plastic deformation behavior and localization prediction of ductile metals which are sensitive to hydrostatic stress and anisotropically damaged. The model is based on a generalized macroscopic theory within the framework of nonlinear continuum damage mechanics. The formulation relies on a multiplicative decomposition of the metric transformation tensor into elastic and damaged-plastic parts. Furthermore, undamaged configurations are introduced which are related to the damaged configurations via associated metric transformations which allow for the interpretation as damage tensors. Strain rates are shown to be additively decomposed into elastic, plastic and damage strain rate tensors. Moreover, based on the standard dissipative material approach the constitutive framework is completed by different stress tensors, a yield criterion and a separate damage condition as well as corresponding potential functions. The evolution laws for plastic and damage strain rates are discussed in some detail. Estimates of the stress and strain histories are obtained via an explicit integration procedure which employs an inelastic (damage-plastic) predictor followed by an elastic corrector step. Numerical simulations of the elastic–plastic deformation behavior of damaged solids demonstrate the efficiency of the formulation. A variety of large strain elastic–plastic-damage problems including severe localization is presented, and the influence of different model parameters on the deformation and localization prediction of ductile metals is discussed.

Finite element implementation of non-linear elastoplastic constitutive laws using local and global explicit algorithms with automatic error control

Implicit stress integration algorithms have been demonstrated to provide a robust formulation for finite element analyses in computational mechanics, but are difficult and impractical to apply to increasingly complex non-linear constitutive laws. This paper discusses the performance of fully explicit local and global algorithms with automatic error control used to integrate general non-linear constitutive laws into a non-linear finite element computer code. The local explicit stress integration procedure falls under the category of return mapping algorithm with standard operator split and does not require the determination of initial yield or the use of any form of stress adjustment to prevent drift from the yield surface. The global equations are solved using an explicit load stepping with automatic error control algorithm in which the convergence criterion is used to compute automatically the coarse load increment size. The proposed numerical procedure is illustrated here through the implementation of a set of elastoplastic constitutive relations including isotropic and kinematic hardening as well as small strain hysteretic non-linearity. A series of numerical simulations confirm the robustness, accuracy and efficiency of the algorithms at the local and global level. Published in 2001 by John Wiley & Sons, Ltd.

Multibody dynamic simulation of the next generation space telescope using finite elements and fuzzy sets

Multibody dynamic simulations are performed for a large deployable space structure using finite element models and an explicit temporal integration procedure. The structure considered is NASA's light-weight, 8 m-aperture next generation space telescope (NGST). The NGST structure consists of beam components (modeled using beam “super-elements”, each composed of two truss elements and a torsional spring), thin-surface shell-type components (modeled using super-elements, each composed of one brick, twelve truss elements, and six surface elements), revolute and prismatic joints, and reaction wheels. Detailed numerical simulations are conducted for the vibrational response, attitude control, and deployment. A fuzzy-set technique is used to assess the effect of changing the dwell time of the various deployment actuators on the total strain energy of the structure.

Large strain elastic-plastic theory and nonlinear finite element analysis based on metric transformation tensors

The present paper is concerned with an efficient framework for a nonlinear finite element procedure for the rate-independent finite strain analysis of solids undergoing large elastic-plastic deformations. The formulation relies on the introduction of a mixed-variant metric transformation tensor which will be multiplicatively decomposed into a plastic and an elastic part. This leads to the definition of an appropriate logarithmic strain measure whose rate is shown to be additively decomposed into elastic and plastic strain rate tensors. The mixed-variant logarithmic elastic strain tensor provides a basis for the definition of a local isotropic hyperelastic stress response in the elastic-plastic solid. Additionally, the plastic material behavior is assumed to be governed by a generalized J2 yield criterion and rate-independent isochoric plastic strain rates are computed using an associated flow rule. On the numerical side, the computation of the logarithmic strain tensors is based on 1st and higher order Pade approximations. Estimates of the stress and strain histories are obtained via a highly stable and accurate explicit scalar integration procedure which employs a plastic predictor followed by an elastic corrector step. The development of a consistent elastic-plastic tangent operator as well as its implementation into a nonlinear finite element program will also be discussed. Finally, the numerical solution of finite strain elastic-plastic problems is presented to demonstrate the efficiency of the algorithm.

Surface Evolution During Semiconductor Processing

We discuss our approach to using the Riemann problem to compute surface profile evolution during the simulation of deposition, etch and reflow processes. Each pair of segments which represents the surface is processed sequentially. For cases in which both segments are the same material, the Riemann problem is solved. For cases in which the two segments are different materials, two Riemann problems are solved. The material boundary is treated as the right segment for the left material and as the left segment for the right material. The critical equations for the analyses are the characteristics of the Riemann problem and the ‘jump conditions’ which represent continuity of the surface. Examples are presented to demonstrate selected situations. One limitation of the approach is that the velocity of the surface is not known as a function of the surface angle. Rather, it is known for the angles of the left and right segments. The rate as a function of angle must be assumed for the explicit integration procedure used. Numerical implementation is briefly discussed.

Three dimensional analysis of dynamic ductile crack growth in a thin plate

A three dimensional analysis of dynamic ductile crack growth in an edge cracked plate is carried out, using a data parallel implementation in a transient three dimensional finite element program. An elastic-viscoplastic constitutive relation for a porous plastic solid is used to model ductile fracture by the nucleation and subsequent growth of voids to coalescence. Two populations of second phase particles are represented, large inclusions with low strength, which result in large voids near the crack tip at an early stage, and small second phase particles, which require large strains before cavities nucleate. Adiabatic heating owing to plastic dissipation and the resulting thermal softening are accounted for in the analyses. The crack speed and the crack path are based on the ductile failure predictions of the material model, so that the present study is free from ad hoc assumptions regarding appropriate dynamic crack growth criteria. A convected coordinate Lagrangian formulation is employed and the discretization is based on twenty-node brick elements with 2 × 2 × 2 Gauss points. The equations of motion are integrated numerically by an explicit integration procedure using a lumped mass matrix. Crack growth occurs by tunneling in the central part of the plate and shear lip formation at the free surface. The effect of various material parameters and of plate thickness on this process is studied. For comparison purposes, a calculation with overall plane strain boundary conditions is carried out.

3D analysis of failure modes in the Charpy impact test

A three-dimensional analysis of the Charpy V-notch specimen subject to impact loading is carried out, using a data parallel implementation in a transient three-dimensional finite element programme. An elastic-viscoplastic constitutive relation for a porous plastic solid is used to model ductile fracture by the nucleation and subsequent growth of voids to coalescence. Cleavage is modelled in terms of attaining a critical value of the maximum principal normal stress over a specified material volume. A convected coordinate Lagrangian formulation is employed and the discretization is based on twenty-node brick elements with 2*2*2 Gauss points. The equations of motion are integrated numerically by an explicit integration procedure using a lumped mass matrix. The predictions of the full three-dimensional calculations are compared with those of corresponding plane-strain analyses.

An Advanced Method for Transient Temperature Calculation in Fuel Element Structural Analysis

An advanced method has been developed for the specific purpose of calculating temperatures in fuel element structural analysis. Fuel, cladding, coolant, and structural temperatures are treated by a single system of equations. Melting of the fuel and cladding and boiling of the coolant are included in the model. The method is compared to other solution techniques. The thermal characteristics of the finite element method (FEM) and finite difference method (FDM) transient calculations are compared. The present method includes FDM and FEM algorithms as special cases; an optimum combination of both techniques is the standard usage. Explicit, implicit, or Crank-Nicholson integration procedures are possible. The method is fast running, reliable, and has no stability problems. The new method has been implemented into the temperature calculation subcode system TEMPER for use with URANUS or other fuel element codes. Special attention has been given to user requirements (e.g., an automatic time-step control). The URANUS code, with this subcode system TEMPER, has been applied successfully to difficult fast breeder fuel rod analysis including transient overpower, loss of flow, local coolant blockage, and specific carbide fuel experiments.