H.S. Yu, S.W. Sloan and P.W. Kleeman
This paper presents a new finite element formulation of the upper bound theorem. The formulation uses a six‐noded linear strain triangular element. Each node has two unknown…
Abstract
This paper presents a new finite element formulation of the upper bound theorem. The formulation uses a six‐noded linear strain triangular element. Each node has two unknown velocities and each corner of a triangle is associated with a specified number of unknown plastic multiplier rates. The major advantage of using a linear strain element, rather than a constant strain element, is that the velocity field can be modelled more accurately. In addition, the incompressibility condition can be easily satisfied without resorting to special arrangements of elements in the mesh. The formulation permits kinematically admissible velocity discontinuities at specified locations within the finite element mesh. To ensure that finite element formulation of the upper bound theorem leads to a linear programming problem, the yield criterion is expressed as a linear function of the stresses. The linearized yield surface is defined to circumscribe the parent yield surface so that the solution obtained is a rigorous upper bound. During the solution phase, an active set algorithm is used to solve the resulting linear programming problem. Several numerical examples are given to illustrate the capability of the new procedure for computing rigorous upper bounds. The efficiency and accuracy of the quadratic formulation is compared with that of the 3‐noded constant strain formulation in detail.
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S.R.P. MEDEIROS, P.M. PIMENTA and P. GOLDENBERG
A new algorithm for reducing the profile and root‐mean‐square wavefront of sparse matrices with a symmetric structure is presented. Our numerical experiments show an overall…
Abstract
A new algorithm for reducing the profile and root‐mean‐square wavefront of sparse matrices with a symmetric structure is presented. Our numerical experiments show an overall better performance than the widely used reverse Cuthill‐McKee, Gibbs‐King and Sloan algorithms. The new algorithm is fast, simple and useful in engineering analysis where it can be employed to derive efficient orderings for both profile and frontal solution schemes.
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Scott W. Sloan, Andrew J. Abbo and Daichao Sheng
Effective explicit algorithms for integrating complex elastoplastic constitutive models, such as those belonging to the Cam clay family, are described. These automatically divide…
Abstract
Effective explicit algorithms for integrating complex elastoplastic constitutive models, such as those belonging to the Cam clay family, are described. These automatically divide the applied strain increment into subincrements using an estimate of the local error and attempt to control the global integration error in the stresses. For a given scheme, the number of substeps used is a function of the error tolerance specified, the magnitude of the imposed strain increment, and the non‐linearity of the constitutive relations. The algorithms build on the work of Sloan in 1987 but include a number of important enhancements. The steps required to implement the integration schemes are described in detail and results are presented for a rigid footing resting on a layer of Tresca, Mohr‐Coulomb, modified Cam clay and generalised Cam clay soil. Explicit methods with automatic substepping and error control are shown to be reliable and efficient for these models. Moreover, for a given load path, they are able to control the global integration error in the stresses to lie near a specified tolerance. The methods described can be used for exceedingly complex constitutive laws, including those with a non‐linear elastic response inside the yield surface. This is because most of the code required to program them is independent of the precise form of the stress‐strain relations. In contrast, most of the implicit methods, such as the backward Euler return scheme, are difficult to implement for all but the simplest soil models.
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Pedro Miguel de Almeida Areias, Timon Rabczuk and Joaquim Infante Barbosa
– The purpose of this paper is to discuss the linear solution of equality constrained problems by using the Frontal solution method without explicit assembling.
Abstract
Purpose
The purpose of this paper is to discuss the linear solution of equality constrained problems by using the Frontal solution method without explicit assembling.
Design/methodology/approach
Re-written frontal solution method with a priori pivot and front sequence. OpenMP parallelization, nearly linear (in elimination and substitution) up to 40 threads. Constraints enforced at the local assembling stage.
Findings
When compared with both standard sparse solvers and classical frontal implementations, memory requirements and code size are significantly reduced.
Research limitations/implications
Large, non-linear problems with constraints typically make use of the Newton method with Lagrange multipliers. In the context of the solution of problems with large number of constraints, the matrix transformation methods (MTM) are often more cost-effective. The paper presents a complete solution, with topological ordering, for this problem.
Practical implications
A complete software package in Fortran 2003 is described. Examples of clique-based problems are shown with large systems solved in core.
Social implications
More realistic non-linear problems can be solved with this Frontal code at the core of the Newton method.
Originality/value
Use of topological ordering of constraints. A-priori pivot and front sequences. No need for symbolic assembling. Constraints treated at the core of the Frontal solver. Use of OpenMP in the main Frontal loop, now quantified. Availability of Software.
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G.P. Nikishkov, A. Makinouchi, G. Yagawa and S. Yoshimura
An algorithm for domain partitioning with iterative load balancing is presented. A recursive graph labeling scheme is used to distribute elements among subdomains at each…
Abstract
An algorithm for domain partitioning with iterative load balancing is presented. A recursive graph labeling scheme is used to distribute elements among subdomains at each iteration. Both graph distance information and information about neighbor vertices are employed during the labeling process. Element quantities for balanced subdomains are predicted, solving the algebraic load balancing problem after each iteration. The same graph labeling scheme with slight modifications is applied to node renumbering inside subdomains. The proposed algorithm is especially suitable for load balancing when a direct method is used for subdomain condensation and the evaluation of cost function is time consuming. Several examples of optimized partitioning of irregular and regular meshes show that load balancing can be achieved with one to three iterations.
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Luying Ju, Zihai Yan, Mingming Wu, Gangping Zhang, Jiajia Yan, Tianci Yu, Pan Ding and Riqing Xu
The purpose of this paper is to suggest an implicit integration method for updating the constitutive relationships in the newly proposed anisotropic egg-shaped elastoplastic…
Abstract
Purpose
The purpose of this paper is to suggest an implicit integration method for updating the constitutive relationships in the newly proposed anisotropic egg-shaped elastoplastic (AESE) model and to apply it in ABAQUS.
Design/methodology/approach
The implicit integration algorithm based on the Newton–Raphson method and the closest point projection scheme containing an elastic predictor and plastic corrector are implemented in the AESE model. Then, the integration code for this model is incorporated into the commercial finite element software ABAQUS through the user material subroutine (UMAT) interface to simulate undrained monotonic triaxial tests for various saturated soft clays under different consolidation conditions.
Findings
The comparison between the simulated results from ABAQUS and the experimental results demonstrates the satisfactory performance of this implicit integration algorithm in terms of effectiveness and robustness and the ability of the proposed model to predict the characteristics of soft clay.
Research limitations/implications
The rotational hardening rule in the AESE model together with the implicit integration algorithm cannot be considered.
Originality/value
The singularity problem existing in most elastoplastic models is eliminated by the closed, smooth and flexible anisotropic egg-shaped yield surface form in the AESE model. In addition, this notion leads to an efficient implicit integration algorithm for updating the highly nonlinear constitutive equations for unsaturated soft clay.
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The initial stiffness method has been extensively adopted for elasto‐plastic finite element analysis. The main problem associated with the initial stiffness method, however, is…
Abstract
Purpose
The initial stiffness method has been extensively adopted for elasto‐plastic finite element analysis. The main problem associated with the initial stiffness method, however, is its slow convergence, even when it is used in conjunction with acceleration techniques. The Newton‐Raphson method has a rapid convergence rate, but its implementation resorts to non‐symmetric linear solvers, and hence the memory requirement may be high. The purpose of this paper is to develop more advanced solution techniques which may overcome the above problems associated with the initial stiffness method and the Newton‐Raphson method.
Design/methodology/approach
In this work, the accelerated symmetric stiffness matrix methods, which cover the accelerated initial stiffness methods as special cases, are proposed for non‐associated plasticity. Within the computational framework for the accelerated symmetric stiffness matrix techniques, some symmetric stiffness matrix candidates are investigated and evaluated.
Findings
Numerical results indicate that for the accelerated symmetric stiffness methods, the elasto‐plastic constitutive matrix, which is constructed by mapping the yield surface of the equivalent material to the plastic potential surface, appears to be appealing. Even when combined with the Krylov iterative solver using a loose convergence criterion, they may still provide good nonlinear convergence rates.
Originality/value
Compared to the work by Sloan et al., the novelty of this study is that a symmetric stiffness matrix is proposed to be used in conjunction with acceleration schemes and it is shown to be more appealing; it is assembled from the elasto‐plastic constitutive matrix by mapping the yield surface of the equivalent material to the plastic potential surface. The advantage of combining the proposed accelerated symmetric stiffness techniques with the Krylov subspace iterative methods for large‐scale applications is also emphasized.
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The paper describes the derivation and application of a range of numerical algorithms for implementing the Mohr—Coulomb yield criterion in a non‐linear finite element computer…
Abstract
The paper describes the derivation and application of a range of numerical algorithms for implementing the Mohr—Coulomb yield criterion in a non‐linear finite element computer program. Emphasis is placed on the difficulties associated with the corners of the yield surface. In contrast to the more conventional forward‐Euler procedures, a backward‐Euler integration technique is adopted. A range of methods, including a ‘consistent approach’ are used to derive the tangent modular matrix. Numerical experiments are presented which involve solution algorithms including the modified and full Newton—Raphson procedures, ‘line‐searches’ and the arc‐length method. It is shown that the introduction of efficient integration and tangency algorithms can lead to very substantial improvements in the convergence characteristics.
Miroslav Halilovic, Bojan Starman, Marko Vrh and Boris Stok
The purpose of this study, which is designed for the implementation of models in the implicit finite element framework, is to propose a robust, stable and efficient explicit…
Abstract
Purpose
The purpose of this study, which is designed for the implementation of models in the implicit finite element framework, is to propose a robust, stable and efficient explicit integration algorithm for rate-independent elasto-plastic constitutive models.
Design/methodology/approach
The proposed automatic substepping algorithm is founded on an explicit integration scheme. The estimation of the maximal subincrement size is based on the stability analysis.
Findings
In contrast to other explicit substepping schemes, the algorithm is self-correcting by definition and generates no cumulative drift. Although the integration proceeds with maximal possible subincrements, high level of accuracy is attained. Algorithmic tangent stiffness is calculated in explicit form and optionally no analytical second-order derivatives are needed.
Research limitations/implications
The algorithm is convenient for elasto-plastic constitutive models, described with an algebraic constraint and a set of differential equations. This covers a large family of materials in the field of metal plasticity, damage mechanics, etc. However, it cannot be directly used for a general material model, because the presented algorithm is convenient for solving a set of equations of a particular type.
Practical implications
The estimation of the maximal stable subincrement size is computationally cheap. All expressions in the algorithm are in explicit form, thus the implementation is simple and straightforward. The overall performance of the approach (i.e. accuracy, time consumption) is fully comparable with a default (built-in) ABAQUS/Standard algorithm.
Originality/value
The estimated maximal subincrement size enables the algorithm to be stable by definition. Subincrements are much larger than those in conventional substepping algorithms. No error control, error correction or local iterations are required even in the case of large increments.
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Divyanshu Kumar Lal and Arghya Das
Semi-implicit type cutting plane method (CPM) and fully implicit type closest point projection method (CPPM) are the two most widely used frameworks for numerical stress…
Abstract
Purpose
Semi-implicit type cutting plane method (CPM) and fully implicit type closest point projection method (CPPM) are the two most widely used frameworks for numerical stress integration. CPM is simple, easy to implement and accurate up to first order. CPPM is unconditionally stable and accurate up to second order though the formulation is complex. Therefore, this study aims to develop a less complex and accurate stress integration method for complex constitutive models.
Design/methodology/approach
Two integration techniques are formulated using the midpoint and Romberg method by modifying CPM. The algorithms are implemented for three different classes of soil constitutive model. The efficiency of the algorithms is judged via stress point analysis and solving a boundary value problem.
Findings
Stress point analysis indicates that the proposed algorithms are stable even with a large step size. In addition, numerical analysis for solving boundary value problem demonstrates a significant reduction in central processing unit (CPU) time with the use of the semi-implicit-type midpoint algorithm.
Originality/value
Traditionally, midpoint and Romberg algorithms are formulated from explicit integration techniques, whereas the present study uses a semi-implicit approach to enhance stability. In addition, the proposed stress integration algorithms provide an efficient means to solve boundary value problems pertaining to geotechnical engineering.