Hybrid-mixed Stress Research Articles

Non-linear dynamic analysis of reinforced concrete structures with hybrid mixed stress finite elements

This paper presents and discusses a hybrid-mixed stress (HMS) finite element model for the static and dynamic analysis of 2D reinforced concrete frame structures. It is assumed a physically non-linear behaviour for both concrete and steel. In this model, both the stress and the displacement fields are approximated in the domain of each element. Complete sets of orthonormal Legendre polynomials are used as approximation functions. The use of these functions makes possible the definition of analytical closed form solutions for the computation of all structural operators when a physically linear behaviour is assumed and leads to the development of very effective p- refinement procedures. Due to the nature of the HMS formulation, it is possible to consider the use of both inverse and direct integration schemes at the cross-section level. The time integration procedure uses classical step-by-step techniques, such as Newmark and α-HHT method. The latter is used in order to supress some numerical noise that may occur in the iterative procedure involved in the solution of dynamic non-linear problems. Three different types of damage models, with and without permanent strains, are used to model concrete behaviour. To validate the model, to illustrate its potential and to assess its accuracy and efficiency, several numerical examples are discussed, and comparisons are made with solutions provided by other numerical techniques.

Three-dimensional hybrid-mixed stress elements for free vibration analysis

Hexahedral hybrid-mixed finite elements are proposed for free vibration analysis of three-dimensional solids. The element formulation relies on the simultaneous and independent approximations of stress and displacement in the element domain as well as the displacement on their boundary. Sets of complete and linearly independent non-nodal Legendre polynomials used for the field variables lead to symmetric, highly sparse and well conditioned solving systems. Numerical tests show that the elements yield accurate results, even in the presence of high stress gradients, and they seem to be free of shear and volumetric locking and have low sensitivity to mesh distortion. The hierarchical p-refinement strategy is exploited.

On the parallel implementation of a hybrid-mixed stress formulation

This paper reports the parallel implementation of a hybrid-mixed stress finite element model for the analysis of three-dimensional structures. The stress and the displacement fields in the domain and the displacement fields on the static boundary are independently approximated. All field equations are imposed in a weighted residual form and Legendre polynomials are used to define the approximation bases. The parallel implementation of the model uses a message passing scheme based on the MPI standard. For the parallel solution of the governing system, two distinct approaches are tested. The first corresponds to the use of a third-party parallel direct solver code, named MUMPS. The second uses an in-house developed hybrid solver inspired by the FETI method and designed to deal with the typical pattern of the hybrid-mixed stress model governing system. Some numerical examples are discussed to illustrate the effectiveness and performance of the adopted parallel approach.

Computation of critical loads and buckling modes using hybrid-mixed stress finite element models

This work presents a hybrid mixed stress finite element model for the linear stability analysis of frame structures and Reissner–Mindlin plate bending problems. This model approximates the stress and displacement fields in the domain and the displacement field on the static boundary. Legendre polynomials are used to define all approximation bases, in order to compute analytical solutions for all structural operators. It enables also the implementation of highly efficient p-refinement procedures. A material linear behaviour is assumed and to validate the model and to illustrate its potential, numerical tests are presented and compared with analytical and numerical solutions.

Static and dynamic physically non-linear analysis of concrete structures using a hybrid mixed finite element model

A new hybrid-mixed stress finite element model for the static and dynamic non-linear analysis of concrete structures is presented and discussed in this paper. The main feature of this model is the simultaneous and independent approximation of the stress, the strain and the displacement fields in the domain of each element. The displacements along the static boundary, which is considered to include inter-element boundaries, are also directly approximated. To define the approximation bases in the space domain, complete sets of orthonormal Legendre polynomials are used. The adoption of these functions enables the use of analytical closed form solutions for the computation of all linear structural operators and leads to the development of very effective p-refinement procedures. To represent the material quasi-brittle behaviour, a physically non-linear model is considered by using damage mechanics. A simple isotropic damage model is adopted and to control strain localisation problems a non-local integral formulation is considered. To solve the dynamic non-linear governing system, a time integration procedure based on the use of the @a-HHT method is used. For each time step, the solution of the non-linear governing system is achieved using an iterative algorithm based on a secant method. The model being discussed is applied to the solution of two-dimensional structures. To validate the model, to illustrate its potential and to assess its accuracy and numerical efficiency, several numerical examples are discussed and comparisons are made with solutions provided by experimental tests and with other numerical results obtained using conventional finite elements (CFE).

A new hybrid-mixed stress model for the analysis of concrete structures using damage mechanics

This paper presents a new hybrid-mixed stress finite element model for the static analysis of concrete structures. A non-linear behaviour at the material level is assumed and to model the material’s quasi-brittle behaviour damage mechanics is considered. In this model, the stress, the strain and the displacement fields are independently approximated in the domain of each element. The displacements along the static boundary, which is considered to include inter-element boundaries, are also directly approximated. Complete sets of orthonormal Legendre polynomials are used to define all approximation bases required by the finite element model. The adoption of these functions enables the use of analytical closed form solutions for the computation of all linear structural operators and leads to the development of very effective p-refinement procedures. A simple isotropic damage model is adopted and to control strain localization problems a non-local integral formulation is considered. The model being discussed is applied to the solution of two-dimensional structures. To validate the model, to illustrate its potential and to assess its accuracy and numerical efficiency, several numerical examples are discussed and comparisons are made with solutions provided by experimental tests and with other numerical results obtained using conventional finite elements.

GENERALIZED FINITE ELEMENT METHOD ON NONCONVENTIONAL HYBRID-MIXED FORMULATION

The generalized finite element method (GFEM) is applied to a nonconventional hybrid-mixed stress formulation (HMSF) for plane analysis. In the HMSF, three approximation fields are involved: stresses and displacements in the domain and displacement fields on the static boundary. The GFEM–HMSF shape functions are then generated by the product of a partition of unity associated to each field and the polynomials enrichment functions. In principle, the enrichment can be conducted independently over each of the HMSF approximation fields. However, stability and convergence features of the resulting numerical method can be affected mainly by spurious modes generated when enrichment is arbitrarily applied to the displacement fields. With the aim to efficiently explore the enrichment possibilities, an extension to GFEM–HMSF of the conventional Zienkiewicz-Patch-Test is proposed as a necessary condition to ensure numerical stability. Finally, once the extended Patch-Test is satisfied, some numerical analyses focusing on the selective enrichment over distorted meshes formed by bilinear quadrilateral finite elements are presented, thus showing the performance of the GFEM–HMSF combination.

Structural dynamic analysis using hybrid and mixed finite element models

This paper presents and discusses a hybrid-mixed stress finite element model for the dynamic analysis of structures, assuming a physically and geometrically linear behavior. In this model, both the stress and the displacement fields are approximated in the domain of each element. The displacements along the static boundary are also independently approximated. Orthonormal Legendre polynomials are used as approximation functions. The use of these functions enables the use of analytical closed form solutions for the computation of all structural operators and leads to the development of very effective p-refinement procedures. The linear dynamic analysis is performed using time integration procedures. For this purpose, the classical Newmark, Wilson-θ and α-HHT methods are implemented and tested. A recent and promising alternative approach, based on the definition of a mixed approximation in the time domain is also implemented and assessed. The model being discussed is applied to the solution of plane elasticity and Reissner–Mindlin plate bending problems. To validate the model, to illustrate its potential and to assess its accuracy and efficiency, several numerical examples are discussed and comparisons are made with analytical solutions and solutions provided by other numerical techniques.

Damage analysis of concrete structures using polynomial wavelets

This paper presents and discusses a hybrid-mixed stress finite element model based on the use of polynomial wavelets for the physically non-linear analysis of concrete structures. The effective stress and the displacement fields in the domain of each element and the displacements on the static boundary are independently approximated. As none of the fundamental equations is locally enforced a priori, the hybrid-mixed stress formulation enables the use of a wide range of functions. In the numerical model reported here, all approximations are defined using complete sets of polynomial wavelets. These bases present some important features. In one hand, the functions are orthogonal, which is an important issue when implementing hybrid-mixed stress elements as it ensures high levels of sparsity. On the other hand, the polynomial wavelet basis is defined through linear combinations of Legendre polynomials. This fact enables the use of closed-form solutions for the computation of the integrations involved in the definition of all linear structural operators. A simple isotropic damage model is adopted and a non-local integral formulation where the strain energy release rate is taken as the non-local variable is considered. The numerical model is both incremental and iterative and is solved with a modified Newton–Raphson method that uses the secant matrix. Classical benchmark tests are chosen to illustrate the use of the model under discussion and to assess its numerical performance.

Polynomial wavelets in hybrid‐mixed stress finite element models

AbstractThis paper reports the use of polynomial wavelets as approximation functions in hybrid‐mixed stress finite element models applied to the solution of plane elasticity problems. The stress and displacement fields in the domain and the displacements on the static boundary are independently approximated. The kinematic boundary conditions are locally satisfied. All remaining equations are enforced in a weighted residual form so designed as to ensure that the discrete model embodies the relevant properties of continuum systems, namely the static‐kinematic duality and elastic reciprocity. A set of numerical applications is presented to illustrate the use of the hybrid‐mixed model and to assess its efficiency and accuracy. Copyright © 2009 John Wiley & Sons, Ltd.

Continuum damage models with non-conventional finite element formulations

In recent years, some research effort has been devoted to the development of non-conventional finite element models for the analysis of concrete structures. These models use continuum damage mechanics to represent the physically non-linear behavior of this quasi-brittle material. Two alternative approaches proved to be robust and computationally competitive when compared with the classical displacement finite element implementations. The first corresponds to the hybrid–mixed stress model where both the effective stress and the displacement fields are independently modeled in the domain of each finite element and the displacements are approximated along the static boundary, which is considered to include the inter-element edges. The second approach corresponds to a hybrid–displacement model. In this case, the displacements in the domain of each element and the tractions along the kinematic boundary are independently approximated. Since it is a displacement model, the inter-element boundaries are now included in the kinematic boundary. In both models, complete sets of orthonormal Legendre polynomials are used to define all approximations required, so very effective p-refinement procedures can be implemented. This paper illustrates the numerical performance of these two alternative approaches and compares their efficiency and accuracy with the classical finite element models. For this purpose, a set of numerical tests is presented and discussed.

Performance of Walsh-based hybrid-mixed stress analysis

The most relevant aspects involved in the implementation of the stress model of the hybrid-mixed finite element formulation are presented and discussed. In this formulation, the stress and displacement fields in the domain of the element and the displacements on the boundary are simultaneously approximated. Digital Walsh functions are used in all three approximations. Due to the special properties of these functions, it is possible to obtain closed form solutions for the integrals involved in the computation of the structural operators. The governing system is usually very large in dimension, but also well structured and highly sparse. Consequently, the use of adequate algorithms to store and manipulate sparse matrices is essential to ensure the numerical efficiency of the model. The direct and iterative methods most used in the solution of large, sparse systems of linear equations are tested and assessed. Fast transform algorithms are used in the post-processing phase to perform the linear combinations of digital Walsh functions needed to construct the stress and displacement fields from their direct approximations. Linear elastostatic problems are used to illustrate the implementation of the Walsh-based hybrid-mixed stress elements.

Hybrid-mixed stress finite element models in elastoplastic analysis

A hybrid-mixed stress finite element formulation for the elastoplastic analysis of stretching plates is presented. This model is characterized by the simultaneous and independent approximation of both the stress and the displacement fields in the domain and of the displacement field on the static boundary. To model the local phenomena associated with plasticity, the plastic parameter increments are also directly approximated. The plastic flow and the kinematic boundary conditions are locally satisfied. The remaining fundamental equations are enforced in a weighted residual form so designed as to ensure that the discrete model presents all relevant properties of the continuous system, namely the static–kinematic duality, elastic reciprocity and associated plasticity. The orthogonal Legendre polynomials are used as approximation functions for the stress and the displacements fields. Dirac functions and non-negative polynomial functions are used to model the plastic parameter increments. The model presented here assumes a quasi-static and geometrically linear response. The elastoplastic constitutive relations are uncoupled into elastic and plastic deformation modes and both the Von Mises and the Drucker–Prager yield criteria have been implemented. The non-linear governing system is solved using the Newton–Raphson method. To validate the hybrid-mixed stress model and to assess its performance and accuracy, a set of numerical test cases is presented and discussed.

Mixed and hybrid stress elements for biphasic media

The hybrid-mixed stress element for transient analysis of compressible and incompressible biphasic media is based on the approximation of the stress and pressure fields in the domain of the element and of the displacements in the solid and fluid phases independently in the domain and on the boundary. The hybrid and hybrid-Trefftz variants are derived constraining the domain approximation to satisfy locally the equilibrium condition and all domain conditions, respectively. As it is typical of stress elements, in these alternative formulations the inter-element and boundary continuity conditions are enforced on the surface forces acting on the solid and fluid phases of the medium. The hybrid-Trefftz stress element is applied to the transient analysis of two-dimensional and axisymmetric biphasic media. The performance of the element is illustrated in terms of sensitivity to distortion and to wavelength, full incompressibility and spectral and transient modelling of both bounded and unbounded domains.

Implementation of an hybrid-mixed stress model based on the use of wavelets

This paper illustrates the use of wavelets defined on the interval as approximation functions in hybrid-mixed stress finite element models applied to the solution of linear elastic plate stretching and plate bending problems. Special attention is given to the algorithms and techniques involved in the generation and manipulation of such functions. A set of numerical examples is presented to illustrate the use of the hybrid-mixed model.

Hybrid-mixed stress model for the nonlinear analysis of concrete structures

This communication reports on the nonlinear analysis of 2D concrete structures using an hybrid-mixed stress formulation and continuum damage mechanics. The finite element model adopted is based on the direct and independent approximation of three fields: the stress and displacement fields in the domain of each element and the displacement field on the static boundary. The nonlinear behavior of the material is modelled using both a local and nonlocal isotropic damage model. Two numerical examples are presented to illustrate and validate the use of such numerical technique.

Wavelets in hybrid-mixed stress elements

The objective of this paper is to present the application of wavelets in the implementation of the stress model of the hybrid-mixed finite element formulation. Independent wavelet bases are used to approximate the stresses in the domain and the displacements both in the domain and on the boundary of elastostatic finite elements. Except for the kinematic boundary conditions, which are enforced directly, all the remaining equations of elastostatics are enforced in a weighted residual form so designed as to ensure that the discrete model embodies the relevant properties of the continuum fields they represent, namely static-kinematic duality and elastic reciprocity. Numerical examples are used to illustrate the characteristics of the numerical model being presented and to assess its accuracy and efficiency.