Material Nonlinear Problems Research Articles

Static analysis using flexibility disassembly perturbation for material nonlinear problem with uncertainty

Nonlinear finite element analysis of large-scale structures usually requires a lot of calculation cost, because it is necessary to repeatedly inverse the modified stiffness matrix caused by nonlinearity in the calculation process. When considering the uncertainty in materials, the calculation of the nonlinear analysis will be more unbearable. To improve the computational efficiency, this work develops a new method for nonlinear analysis with material uncertainty based on flexibility disassembly perturbation (FDP) approach. The FDP is an algorithm that can quickly calculate the inverse of a stiffness matrix. The basic idea of the proposed method is to introduce the FDP formula into Newton-Raphson iteration method to accelerate the nonlinear iterative calculation. Three numerical examples, one statically determinate structure and two statically indeterminate structures, are used to verify the accuracy and efficiency of the proposed method. The results show that the calculation time of the proposed method is far less than that of the existing complete analysis and combined approximation algorithms. In terms of computational accuracy, for statically determinate structures, the proposed algorithm can obtain exact solutions that are identical to the complete analysis results, while for statically indeterminate structures, the proposed algorithm can obtain approximate solutions that are very close to the complete analysis results.

Adaptive Load Incremental Step in Large Increment Method for Elastoplastic Problems

As a force-based finite element method (FEM), large increment method (LIM) shows unique advantages in material nonlinearity problems. In LIM for material nonlinearity analysis, adaptive load incremental step is a fundamental step for its successful application. In this work, a strategy to automatically refine the load incremental step is proposed in the framework of LIM. The adaptive load incremental step is an iterative process based on the whole loading process, and the location and number of post-refined incremental steps are determined by the posteriori error of energy on the pre-refined incremental steps. Furthermore, the iterative results from the pre-refined incremental steps can be utilized as the initial value to calculate the result for the post-refined incremental steps, which would significantly improve the computational accuracy and efficiency. The strategy is demonstrated using a two-dimensional example with a bilinear hardening material model under cyclic loading, which verifies the accuracy and efficiency of the strategy in LIM. Compared with the displacement-based FEM, which relies upon a step-by-step incremental approach stemming from flow theory, the adaptive load incremental step based on the whole loading process of LIM can avoid the cumulative errors caused by step-by-step in global stage and can quantify the accuracy of results. This work provides a guidance for the practical application of LIM in nonlinear problems.

Parallel Efficiency Analysis of Large Increment Method Based on OpenMP

Large Increment Method (LIM) is a novel force-based method based on the generalized inverse matrix theory for solving material nonlinear problems in small deformation and small displacement conditions. For the spatial parallel computation, LIM can compute the constitutive equations of each element in parallel. By analyzing the parallel-calculating characteristics of LIM, the OpenMP is used for the parallel computation in spatial domain. Through the parallelism analysis of the spatial and time domain for LIM, it shows that LIM has an excellent parallel efficiency in space parallel computation.

PolyStress: a Matlab implementation for local stress-constrained topology optimization using the augmented Lagrangian method

We present PolyStress, a Matlab implementation for topology optimization with local stress constraints considering linear and material nonlinear problems. The implementation of PolyStress is built upon PolyTop, an educational code for compliance minimization on unstructured polygonal finite elements. To solve the nonlinear elasticity problem, we implement a Newton-Raphson scheme, which can handle nonlinear material models with a given strain energy density function. To solve the stress-constrained problem, we adopt a scheme based on the augmented Lagrangian method, which treats the problem consistently with the local definition of stress without employing traditional constraint aggregation techniques. The paper discusses several theoretical aspects of the stress-constrained problem, including details of the augmented Lagrangian-based approach implemented herein. In addition, the paper presents details of the Matlab implementation of PolyStress, which is provided as electronic supplementary material. We present several numerical examples to demonstrate the capabilities of PolyStress to solve stress-constrained topology optimization problems and to illustrate its modularity to accommodate any nonlinear material model. Six appendices supplement the paper. In particular, the first appendix presents a library of benchmark examples, which are described in detail and can be explored beyond the scope of the present work.

Topology optimization considering the Drucker–Prager criterion with a surrogate nonlinear elastic constitutive model

We address material nonlinear topology optimization problems considering the Drucker–Prager strength criterion by means of a surrogate nonlinear elastic model. The nonlinear material model is based on a generalized J2 deformation theory of plasticity. From an algorithmic viewpoint, we consider the topology optimization problem subjected to prescribed energy, which leads to robust convergence in nonlinear problems. The objective function of the optimization problem consists of maximizing the strain energy of the system in equilibrium subjected to a volume constraint. The sensitivity analysis is quite effective and efficient in the sense that there is no extra adjoint equation. In addition, the nonlinear structural equilibrium problem is solved through direct minimization of the structural strain energy using Newton’s method with an inexact line search strategy. Four numerical examples demonstrate features of the proposed nonlinear topology optimization framework considering the Drucker–Prager strength criterion.

Inelastic static and dynamic seismic response assessment of frames with stochastic properties

The article studies the inelastic seismic response assessment of structures with spatially varying material properties via the stochastic finite element method. This is a problem with a complex response function due to stochastic variability and material nonlinearity. Existing formulations either assume linear-elastic behaviour, or use dense models in order to capture both stochasticity and inelasticity. This results to models that require excessive computational resources. The article shows that for material nonlinear problems, the stochastic variability can be taken into consideration using flexibility-based models which allow performing repeated time-history simulations within a reasonable computational time, without loss of accuracy. The interpolation can be adjusted through a user-defined parameter, thus avoiding the need for a dense FE mesh. Contrary to past research, the modelling proposed has many applications in structural engineering, since it allows nonlinear, static or dynamic, simulations of real-scale frame structures with stochastic properties. The proposed method is demonstrated in the framework of the spectral representation method combined with Monte Carlo simulations, but it can be combined with any other method for the description of the stochastic field. A generic single-storey steel frame and a six-storey reinforced concrete frame are considered in order to demonstrate the efficiency of the proposed modelling.

WITHDRAWN: A new approach of material nonlinear finite element analysis

Abstract In this article, a novel approach is proposed for solving material nonlinear analysis problems in which a torsion problem is solved. A torsion specimen was tested in the lab and with the experimental data, shear stress and shear strain are first defined. It is to be noticed that in the basic analysis procedure, linearity assumptions are nowhere used. For example, use of shear modulus is eliminated in the basic formulation. Instead, shear stress function and shear strain functions are used as the material input. These shear stress and shear strain functions are derived from the experimental torsion-rotation data. The methodology gives ideal result, either it is linear or nonlinear range of the loading condition, with much more accuracy than the conventional linear analysis approaches. It could be realized that the methodology could be regarded as stress-based structural analysis procedure.

Collapse‐Pounding Dynamic Responses of Adjacent Frame Structures under Earthquake Action

There are a large number of adjacent buildings in practical engineering application. The structure will collapse and impact the adjacent structures once the weak column is destroyed under seismic action, and, finally, the earthquake damage is aggravated. Material nonlinearity, initial imperfections, and contact problems in the process of collapse‐pounding are considered, and the three‐dimensional calculation model with unequal 8‐story and 6‐story height adjacent frames is established. The dynamic response of adjacent structures caused by collapse‐pounding is investigated when there is a weak column at different positions in the 8‐story frame, and the influence of gap size on the dynamic responses of adjacent structures is discussed. The results show that the impact force is larger when the weak column of the 8‐story frame is close to the top of the 6‐story frame; pounding increases the interlayer displacement angle of the 8‐story frame and decreases the interlayer displacement angle of the 6‐story frame in general; the impact force decreases first, then increases, and after that decreases with the increase of gap size; and the interlayer displacement angle distribution of the 6‐story frame is significantly affected after the collapse‐pounding of the 8‐story frame with a weak column. The collapse‐pounding problem of adjacent buildings under seismic action is very complex, which should be paid enough attention in engineering design and application.

Investigation of a boundary simulation of continuity using the discrete solid element method

The discrete solid element method is an efficient numerical method that simulates the large deformation, strong material nonlinearity, fracture, and dynamic problems of continuity. In the discrete solid element method model, the spring stiffness of the spherical elements on the boundary is different from that inside the discrete solid element method model based on the principle of conservation of energy. The spring stiffness of the spherical elements on the boundary of the discrete solid element method model is shown to have a significant effect on the macroscopic properties. According to the position of the spherical elements on the boundary of the discrete solid element method model, the spherical elements on the boundary are divided into three types, which are spherical elements on the surface position, on the edge position, and on the corner position. To accurately reflect the mechanical behavior of the material, the principle of energy conservation is used to strictly deduce the spring stiffness of the three types of spherical elements on the boundary, and the relationship between the spring stiffness and elastic constants is established. The numerical example shows that the calculation accuracy of the discrete solid element method in modeling the mechanical behavior of continuity is improved after the spring stiffness of the spherical elements on the boundary is revised. In addition, the applications of the discrete solid element method to dynamic buckling of the thin plate and buckling of the cracked thin plate are also given.

New strong formulation for material nonlinear problems based on the particle difference method

In this paper, we propose a strong form method for analyzing material nonlinear problems. The method utilizes the Particle Difference Method (PDM) which is classified as a strong form meshfree method and discretizes the governing equations based on a complete nodal computation without any integral formulation. The conventional strong form meshfree methods could not explicitly handle the nonlinear material model since they are mostly discretized based on the Navier's equation where kinematic variables are unified into displacement. To explicitly treat the nonlinear constitutive equation in the framework of strong formulation, a double derivative approximation is devised, which removes the need for the use of second order derivative approximation. The momentum equation is directly discretized through the double derivative approximation and is linearized by Newton's method to yield an iterative procedure for finding a converged solution. Stresses and internal variables are updated by the return mapping algorithm and efficiency of the iterative procedure is dramatically improved by the algorithmic tangent modulus. The consistency of the double derivative approximation was shown by the reproducing test. The accuracy and robustness of the developed nonlinear procedure were then verified through various inelastic material problems in one and two-dimensions.

Woodbury Approximation Method for Structural Nonlinear Analysis

AbstractAs an exact method, the Woodbury formula is used to solve local material nonlinearity problems in structural analysis, in which the calculation and factorization of the global stiffness mat...

Rheological-dynamical analogy for analysis of vibrations and low cycle fatigue in internally damped inelastic frame structures

This is a study of viscoelastoplastic (VEP) vibrations and their use for the analysis of low cycle fatigue in internally damped inelastic frame structures (IDIFSs). The background of this inelastic theory is presented in the framework of a mathematical-physical analogy between the rheological model and a dynamical model with viscous damping. The rheological-dynamical analogy (RDA) is a type of inelastic analysis, which transforms one category of material non-linear problems to simpler linear dynamical problems using modal analysis. The aim of this paper is to define internal damping based on both the dynamic modulus and modal damping ratios. The idea underlying these approaches is that fatigue damage appears if internal damping is unevenly distributed over the elements of a structure. The residual force method, which requires the use of the finite element method (FEM), is used for the location of damage and derivation of the fatigue damage vector. Finally, the effective force vector is derived from damage mechanics. An analysis of damaged IDIFSs made of reinforced concrete is carried out. It is shown that the RDA, which correlates with the main mechanical properties of the material measured, can improve the prediction of fatigue damage caused by low cycle fatigue.

Quasi static and dynamic inelastic buckling and failure of folded-plate structures by a full-energy finite strip method

A study on how a mathematical material modeling approach named rheological-dynamical analogy (RDA) can be used to predict the quasi static and dynamic inelastic buckling and failure of structures is presented in this paper. An analysis of the uniformly compressed folded-plate structures, made of isotropic materials, is carried out. Two sources of non-linearity, one involving geometrical non-linearity due to large deflection, and the other involving material non-linearity due to inelastic behavior, are analyzed by implementing a full-energy finite strip method (FSM). The material non-linearity is analyzed using the RDA. A very basic continuum damage model with one damage parameter is implemented in conjunction with a mathematical material modeling approach in order to address stiffness reduction due to inelastic behavior. According to the analogy, a very complicated material non-linear problem in the inelastic range of strains is solved as a simple linear dynamic one. The orthotropic constitutive relations are derived and modulus iterative method for the solution of nonlinear equations is presented.

A fast uncertainty analysis for material nonlinear problems based on reanalysis and semi-analytic strategy

A novel semi-analytic method for interval-based uncertainty analysis is suggested to calculate the uncertainty measure index in this study. As nonlinear problems commonly cannot be solved analytically, an analytic interval-based uncertainty analysis method for linear problems is deduced, and an explicit expression of the measure index of uncertainty is given. To extend this form to nonlinear analysis, a semi-analytic uncertainty analysis (SAUA) method is suggested by integrating gradient-based optimization methods and Taylor expansion. Initially, a pseudo optimal solution is obtained. Sequentially, a Taylor expansion is used to calculate the performance function based on the pseudo optimal solution. In this way, the uncertainty measure index can be obtained analytically. Considering the bottleneck of surrogate model, an efficient reanalysis assisted material nonlinear analysis (RAMNA) method is integrated. Thus, the efficiency and accuracy of the suggested method are both enhanced. The SAUA method has been applied to large-scale reliability analysis of vehicle components. The results show that the SAUA is effective and efficient, especially for large-scale problems. Furthermore, the SAUA method can also be extended for other mechanical design, and contribute to shortening the design period.

A 3D PVP co-rotational formulation for large-displacement and small-strain analysis of bi-modulus materials

An efficient computational method is developed for large-displacement and small-strain analysis of 3D bi-modulus materials which are often found in civil, composite and biological engineering. Based on the parametric variational principle (PVP), a unified constitutive equation of 3D bi-modulus materials is proposed to deal with the problem of numerical instability. The small-strain bi-modulus problem is transformed into a standard linear complementarity problem that can be solved easily by the classic Lemke׳s algorithm. By using the co-rotational approach, the local PVP formulation is combined with an existing co-rotational formulation and a new tangent stiffness matrix, including a parametric variable, is derived for geometrically nonlinear analysis. Traditional stress iteration is not required for calculation of the nodal internal force when the Newton–Raphson scheme or Arc-length method is employed to solve the material and geometric nonlinear problem. Convergence of the proposed algorithm is improved greatly in contrast to the traditionally iterative solution. Also, the proposed algorithm is used to simulate a unilateral contact behavior of staggered bio-composites and the effective elasticity modulus of composites is determined accurately.

A Timoshenko finite element straight beam with internal degrees of freedom

SummaryWe present hereafter the formulation of a Timoshenko finite element straight beam with internal degrees of freedom, suitable for nonlinear material problems in geomechanics (e.g., beam type structures and deep pile foundations). Cubic shape functions are used for the transverse displacements and quadratic for the rotations. The element is free of shear locking, and we prove that one element is able to predict the exact tip displacements for any complex distributed loadings and any suitable boundary conditions. After the presentation of the virtual power and the weak form formulations, the construction of the elementary stiffness matrix is detailed. The analytical results of the static condensation method are provided. It is also proven that the element introduced by Friedman and Kosmatka in , with shape functions depending on material properties, is derived from the new beam element. Validation is provided using linear and material nonlinear applications (reinforced concrete column under cyclic loading) in the context of a multifiber beam formulation. Copyright © 2015 John Wiley & Sons, Ltd.

A simple assumed-strain quadrilateral shell element for finite strains and fracture

This work recovers an established technique for improving quadrilateral shell element performance in both out-of-plane and in-plane bending cases using a mixed formulation. A four-field variational principle is established and we relate, at the discrete level, the Lagrange multipliers and secondary right Cauchy---Green field with the displacement and rotation fields. This is the main contribution of this work. High coarse-mesh accuracy is observed for distorted meshes and the robustness is shown to be adequate for crack propagation simulations. A consistent director normalization is performed, as an alternative to our recent spherical interpolation. Covariant metric components are deduced and exact linearization of the shell element is performed. Full assessment of the element is accomplished, showing similar performance to more costly approaches such as enhanced assumed strain. Patch test is satisfied ab-initio and benchmarks present very accurate results. Numerical experimentation for geometrically and material nonlinear problems is presented, as well as one fracture example using our recently proposed cracked edge technique.

Large increment method for elastic and elastoplastic analysis of plates

The displacement-based finite element method (FEM) has become a method of choice for material nonlinear analysis of plates. For material nonlinear problems, the displacement-based FEM relies upon a step-by-step incremental approach and the repetitive computation of the systematic stiffness matrix. These shortcomings lead to the error accumulation and huge computational consumption, which encourage the reconsideration of force-based methods for elastoplastic problems. In this paper, a force-based Large Increment Method (LIM) is employed for the elastoplastic analysis of plates using a force-based 4-node quadrilateral plate element which is based on Mindlin–Reissner plate theory. The consistent elastoplastic flexibility matrix of plate element is derived and implemented to solve elastoplastic plate problems. Two numerical examples are presented to illustrate the mesh convergence of the plate element by solving the linear elastic thin and moderately thick plate problems by comparing with the analytical solutions and displacement-based plate elements. Two simple elastoplastic plate problems are presented to illustrate the accuracy and the computational efficiency of LIM by comparing with the results from the FEM software ABAQUS.

A new numerical method for determining collapse load-carrying capacity of structure made of elasto-plastic material

Determination of collapse load-carrying capacity of elasto-plastic material is very important in designing structure. The problem is commonly solved by elasto-plastic finite element method (FEM). In order to deal with material nonlinear problem involving strain softening problem effectively, a new numerical method-damped Newton method was proposed. The iterative schemes are discussed in detail for pure equilibrium models. In the equilibrium model, the plasticity criterion and the compatibility of the strains are verified, and the strain increment and plastic factor are treated as independent unknowns. To avoid the stiffness matrix being singularity or condition of matrix being ill, a damping factor α was introduced to adjust the value of plastic consistent parameter automatically during the iterations. According to the algorithm, the nonlinear finite element program was complied and its numerical example was calculated. The numerical results indicate that this method converges very fast for both small load steps and large load steps. Compared with those results obtained by analysis and experiment, the predicted ultimate bearing capacity from the proposed method is identical.

Reusing linear finite elements in material and geometrically nonlinear analysis—Application to plane stress problems

This paper describes a computational approach suitable for combined material and geometrically nonlinear analysis by the Finite Element Method. Its main advantage is reuse: once a finite element has been developed with good performance in linear analysis, extension to material and geometrically nonlinear problems is simplified. Extension to geometrically nonlinear problems is enabled by a corotational kinematic description, and that to material nonlinear problems by an optimization-based solution algorithm. The approach thus comprises three ingredients—the development of a high performance linear finite element (for example, using the ANDES concept), a corotational kinematic description, and an optimization algorithm. The main constraint in the application of the corotational formulation is restriction to small deformational displacements. The paper illustrates the realization of the three ingredients on plane stress problems that exhibit elasto-plastic material behavior. Numerical examples are presented to illustrate the effectiveness of the approach. Comparison is made with respect to solutions provided by the commercial nonlinear code ABAQUS as reference.