Hu Washizu Research Articles

Development of quadratic enhanced assumed strain elements for three-dimensional linear elasticity

Enhanced assumed strain (EAS) elements are widely used in the literature to address the issue of locking associated with conventional elements. However, existing literature in the context of three-dimensional (3D) elasticity problems is predominantly restricted to the eight-node linear EAS elements. Thus, existing 3D EAS elements do not exploit the superior performance offered by quadratic elements over linear elements. In the current work, we propose a novel twenty-seven node quadratic EAS element, which, to the best of our knowledge, is the first such attempt in the literature. Additionally, the manuscript also presents a six-node wedge and an eighteen-node wedge EAS element. The proposed elements are derived methodically by investigating the interrelation between the two-field Hellinger–Reissner (HR) and three-field Veubeke–Hu–Washizu (VHW) variational formulations. The robustness and performance of the proposed EAS elements is demonstrated through numerous examples.

A selectively reduced degree basis for efficient mixed nonlinear isogeometric beam formulations with extensible directors

The effect of higher order continuity in the solution field by using NURBS basis function in isogeometric analysis (IGA) is investigated for an efficient mixed finite element formulation for elastostatic beams. It is based on the Hu–Washizu variational principle considering geometrical and material nonlinearities. Here we present a reduced degree of basis functions for the additional fields of the stress resultants and strains of the beam, which are allowed to be discontinuous across elements. This approach turns out to significantly improve the computational efficiency and the accuracy of the results. We consider a beam formulation with extensible directors, where cross-sectional strains are enriched to avoid Poisson locking by an enhanced assumed strain method. In numerical examples, we show the superior per degree-of-freedom accuracy of IGA over conventional finite element analysis, due to the higher order continuity in the displacement field. We further verify the efficient rotational coupling between beams, as well as the path-independence of the results.

Advanced discretization techniques for hyperelastic physics-augmented neural networks

In the present work, advanced spatial and temporal discretization techniques are tailored to hyperelastic physics-augmented neural networks, i.e., neural network based constitutive models which fulfill all relevant mechanical conditions of hyperelasticity by construction. The framework takes into account the structure of neural network-based constitutive models, in particular, that their derivatives are more complex compared to analytical models. The proposed framework allows for convenient mixed Hu–Washizu like finite element formulations applicable to nearly incompressible material behavior. The key feature of this work is a tailored energy–momentum scheme for time discretization, which allows for energy and momentum preserving dynamical simulations. Both the mixed formulation and the energy–momentum discretization are applied in finite element analysis. For this, a hyperelastic physics-augmented neural network model is calibrated to data generated with an analytical potential. In all finite element simulations, the proposed discretization techniques show excellent performance. All of this demonstrates that, from a formal point of view, neural networks are essentially mathematical functions. As such, they can be applied in numerical methods as straightforwardly as analytical constitutive models. Nevertheless, their special structure suggests to tailor advanced discretization methods, to arrive at compact mathematical formulations and convenient implementations.

Reduced representations of assumed fields for Hu–Washizu solid-shell element

Mixed eight-node (hexahedron) solid-shell elements based on the standard or partial version of the three-field Hu–Washizu (HW) functionals are developed for Green strain. Three reduced representations of the assumed stress/strain fields are selected. They improve effectiveness, yet retaining good accuracy and convergence properties. At the outset, the standard HW functional and the assumed stress/strain representations of the 3D solid element B8-15P (Weissman in Int J Numer Methods Eng 39:2337–2361, 1996) are used to derive a solid-shell element with 51 parameters. To eliminate locking, the ANS method is applied to the thickness strain (Betsch and Stein in Commun Numer Methods Eng 11:899–909, 1995) and to the transverse shear strain (Dvorkin and Bathe in Eng Comput 1:77–88, 1984). It is a correct element which, however, yields too large displacements for coarse meshes and trapezoidal through-thickness shapes. To improve the above formulation, the zeta -independent reduced representations of the assumed stress/ strain fields are selected and the transformations to Cartesian components are modified. The thickness strain is enhanced by the EAS method. The element with 35 parameters is derived from the standard/enhanced HW functional, but, to further reduce the assumed fields, partial/enhanced HW functionals are constructed from the 3D potential energy by applying the Lagrange multiplier method only to selected strain components. In the element with 27 parameters, this is applied to the constant in-plane strain and to the transverse shear strain while in the element with 19 parameters, to the constant in-plane strain only.Two other modifications are implemented to enhance the behavior of these elements: (A) the skew coordinates are used in the reduced representations of the in-plane stress/strain (Wisniewski and Turska in Int J Numer Methods Eng 90:506–536, 2012), and (B) the Residual Bending Flexibility correction of the transverse shear stiffness (MacNeal in Comput Struct 8(2):175–183, 1978) is adapted. Finally, the performance of the proposed solid-shell HW elements is demonstrated on several linear and non-linear examples for the linear elastic material and the hyper-elastic material. The proposed elements are compared to each other and to the best existing elements of this class.

A Hu–Washizu variational approach to self-stabilized virtual elements: 2D linear elastostatics

An original, variational formulation of the Virtual Element Method (VEM) is proposed, based on a Hu–Washizu mixed variational statement for 2D linear elastostatics. The proposed variational framework appears to be ideal for the formulation of VEs, whereby compatibility is enforced in a weak sense and the strain model can be prescribed a priori, independently of the unknown displacement model. It is shown how the ensuing freedom in the definition of the strain model can be conveniently exploited for the formulation of self-stabilized and possibly locking-free low order VEs. The superior performances of the VEs formulated within this framework has been verified by application to several numerical tests.

On the second Piola-Kirchhoff and Cauchy stress tensors in nonlinear shells subjected to displacement-dependent loads

In this article, a nonlinear geometrically exact (GeX) hybrid-mixed four-node solid-shell element under non-conservative loading using the sampling surfaces (SaS) method is developed. The term “geometrically exact” means that the parametrization of the middle surface is known and, therefore, the coefficients of the first and second fundamental forms and the Christoffel symbols are taken exactly at element nodes. The SaS formulation is based on the choice of N SaS parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the displacements of these surfaces as fundamental shell unknowns. Such a choice of unknowns with the use of Lagrange polynomials of degree N–1 in the through-thickness distributions of displacements, strains, and stresses leads to an efficient higher-order shell formulation. To develop the hybrid-mixed four-node solid-shell element, which is free of shear and membrane locking, the Hu–Washizu variational principle is utilized. The proposed GeX hybrid-mixed solid-shell element subjected to follower loads shows superior performance in the case of coarse meshes and allows the use of only one load step in most of the considered numerical examples. It is established that the difference between the second Piola-Kirchhoff and Cauchy stresses in some non-conservative problems for isotropic and composite shells undergoing finite rotations can be significant.

Coupling 2D continuum and beam elements: a mixed formulation for avoiding spurious stresses

This paper presents a novel approach to coupling beam and solid elements. Connecting standard beam elements with solid elements results in a situation where the solid part of the model covers the cross-sectional deformation, while at the beam part assumptions only consider the rigid body movement of the cross-section. Therefore, kinematic assumptions do not include cross-section deformations such as warping and contraction. Due to this restriction, spurious stresses occur at the transition zone. To circumvent this problem, this contribution introduces a mixed hybrid transition element based on an extended Hu–Washizu functional. The extension allows the cross-section to contract and warp. Compared with other approaches that include these phenomena, precomputation of the warping function is unnecessary. The present approach considers the cross-sectional deformation using local variables. On the solid interface, there are only displacement degrees of freedom (depending on the solid discretization), whereas on the beam interface, there are two displacement degrees of freedom and one rotation. This allows beam-like parts of the solid model to be replaced with standard beam elements without affecting the overall accuracy of the model. Besides solid–beam coupling, the proposed formulation can also be used to apply beam-like boundary conditions as well as stress resultants to the solid interface. For the sake of simplicity and without restricting generality, the underlying element formulation is introduced for the 2D linear elastic case. Numerical examples demonstrate that the transition element causes no spurious stress distributions on the solid part. The results of full solid models are compared with mixed beam–solid models.

A mixed method for 3D nonlinear elasticity using finite element exterior calculus

AbstractIn this study, a mixed finite element model for 3D nonlinear elasticity using a Hu–Washizu (HW) type variational principle is presented. This mixed variational principle takes the deformed configuration and sections from its cotangent bundle as the input arguments. The critical points of the proposed HW functional enforce compatibility of these sections with the configuration, in addition to mechanical equilibrium and constitutive relations. Using this variational principle we construct a mixed FE approximation that distinguishes a vector from a 1‐form, a feature not commonly found in FE approximations for nonlinear elasticity. This distinction plays a pivotal role in identifying suitable FE spaces for approximating the 1‐forms appearing in the variational principle. These discrete approximations are constructed using ideas borrowed from finite element exterior calculus, which are in turn used to construct a discrete approximation to our HW functional. The discrete equations describing mechanical equilibrium, compatibility, and constitutive rule, are obtained by seeking extremum of the discrete functional with respect to the respective degrees of freedom. The discrete extremum problem is then solved numerically; we use Newton's method for this purpose. This mixed FE technique is then applied to a few benchmark problems wherein conventional displacement based approximations encounter locking and checker boarding. These studies help establish that our mixed FE approximation, which requires no artificial stabilizing terms, is free of these numerical bottlenecks.

A compatible mixed finite element method for large deformation analysis of two‐dimensional compressible solids in spatial configuration

AbstractIn this article, a new mixed finite element formulation is presented for the analysis of two‐dimensional compressible solids in finite strain regime. A three‐field Hu–Washizu functional, with displacement, displacement gradient and stress tensor considered as independent fields, is utilized to develop the formulation in spatial configuration. Certain constraints are imposed on displacement gradient and stress tensor so that they satisfy the required continuity conditions across the boundary of elements. From theoretical standpoint, simplex elements are best suited for the application of continuity constraints. The techniques that are proposed to implement the constraints facilitate their automatic imposition and, hence, they can be regarded as an important feature of the work. Since the exterior calculus provides the basis for the developments presented herein, the relevant topics are discussed within the context of the work. Various technical aspects of the formulation are described in detail. These aspects help to illuminate the mathematical formulation that might seem vague in the first place and, more importantly, they help to provide an efficient implementation for ensuing developments. The performance of the mixed finite element method is studied through benchmark numerical examples and it is compared with other similar elements. It is shown that the element has excellent convergence properties and it is numerically stable, especially for problems where classical first order elements demonstrate stiff or unstable behavior.

An Alternative Electro-Mechanical Finite Formulation for Functionally Graded Graphene-Reinforced Composite Beams with Macro-Fiber Composite Actuator.

With its extraordinary physical properties, graphene is regarded as one of the most attractive reinforcements to enhance the mechanical characteristics of composite materials. However, the existing models in the literature might meet severe challenges in the interlaminar-stress prediction of thick, functionally graded, graphene-reinforced-composite (FG-GRC)-laminated beams that have been integrated with piezoelectric macro-fiber-composite (MFC) actuators under electro-mechanical loadings. If the transverse shear deformations cannot be accurately described, then the mechanical performance of the FG-GRC-laminated beams with MFC actuators will be significantly impacted by the electro-mechanical coupling effect and the sudden change of the material characteristics at the interfaces. Therefore, a new electro-mechanical coupled-beam model with only four independent displacement variables is proposed in this paper. Employing the Hu–Washizu (HW) variational principle, the precision of the transverse shear stresses in regard to the electro-mechanical coupling effect can be improved. Moreover, the second-order derivatives of the in-plane displacement parameters have been removed from the transverse-shear-stress components, which can greatly simplify the finite-element implementation. Thus, based on the proposed electro-mechanical coupled model, a simple C0-type finite-element formulation is developed for the interlaminar shear-stress analysis of thick FG-GRC-laminated beams with MFC actuators. The 3D elasticity solutions and the results obtained from other models are used to assess the performance of the proposed finite-element formulation. Additionally, comprehensive parametric studies are performed on the influences of the graphene volume fraction, distribution pattern, electro-mechanical loading, boundary conditions, lamination scheme and geometrical parameters of the beams on the deformations and stresses of the FG-GRC-laminated beams with MFC actuators.

A variational framework for the strain-smoothed element method

This paper is devoted to a rigorous mathematical foundation for the convergence properties of the strain-smoothed element (SSE) method. The SSE method has demonstrated improved convergence behaviors compared to other strain smoothing methods through various numerical examples; however, there has been no theoretical evidence for the convergence behavior. A unique feature of the SSE method is the construction of smoothed strain fields within elements by fully unifying the strains of adjacent elements. Owing to this feature, convergence analysis is required, which is different from other existing strain smoothing methods. In this paper, we first propose a novel mixed variational principle wherein the SSE method can be interpreted as a Galerkin approximation of that. The proposed variational principle is a generalization of the well-known Hu–Washizu variational principle; thus, various existing strain smoothing methods can be expressed in terms of the proposed variational principle. With a unified view of the SSE method and other existing methods through the proposed variational principle, we analyze the convergence behavior of the SSE method and explain the reason for the improved performance compared to other methods. We also present numerical experiments that support our theoretical results.

Coupled thermoelectroelastic analysis of thick and thin laminated piezoelectric structures by exact geometry solid‐shell elements based on the sampling surfaces method

AbstractA hybrid‐mixed exact geometry four‐node thermopiezoelectric solid‐shell element through the sampling surfaces (SaS) formulation is proposed. The SaS formulation is based on the choice of an arbitrary number of SaS within layers parallel to the middle surface and located at Chebyshev polynomial nodes in order to introduce the temperatures, displacements and electric potentials of these surfaces as basic shell unknowns. Due to the variational formulation, the outer surfaces and interfaces are also included into a set of SaS. Such choice of unknowns with the use of Lagrange polynomials in the through‐thickness approximations of temperatures, temperature gradient, displacements, strains, electric potential, and electric field leads to a very compact higher‐order thermopiezoelectric shell formulation. To implement efficient analytical integration throughout the solid‐shell element, the extended assumed natural strain method is employed for all components of the temperature gradient, strain tensor, and electric field vector. The developed hybrid‐mixed four‐node thermopiezoelectric solid‐shell element is based on the Hu–Washizu variational principle and shows a superior performance in the case of coarse meshes. It can be useful for the three‐dimensional thermoelectroelastic analysis of thick and thin doubly curved laminated piezoelectric shells under thermal loading, since the SaS formulation allows one to obtain the numerical solutions with a prescribed accuracy, which asymptotically approach exact solutions of the theory of thermopiezoelectricity as a number of SaS tends to infinity.

Hybrid-Mixed Low-Order Finite Elements for Geometrically Exact Shell Models: Overview and Comparison

We critically review and compare fifteen mixed and mixed-hybrid nonlinear shell finite element formulations with 4 nodes in order to identify those that are closest to the “optimal” one. We consider formulations that are based either on Assumed Natural Strain concept, Enhanced Assumed Strain concept, Hellinger–Reissner functional or Hu–Washizu functional, and those that effectively combine several of the mentioned approaches. Most of the formulations are state-of-the-art, but some are also presented for the first time. We show that the flexibility of mixed-hybrid formulations requires careful consideration of theoretical and numerical aspects in the process of design of a high-performance element. We make extensive numerical experiments in order to assess convergence properties, mesh distortion sensitivity, and computational speed (that is associated with the ability to achieve large steps during the solution search) of the considered formulations. We show that the application of the idea to combine mixed-hybrid formulations of the Hellinger–Reissner and Hu–Washizu type with the Assumed Natural Strain and/or the Enhanced Assumed Strain concepts generates the formulations that perform excellent. We also identify the formulations that are currently closest to the “optimal” one.

Stability of Layered Structures with Hybridized Configuration by Means of a Reddy-Type Higher-Order Finite Element Formulation

To make use of the merit of designability, each lamina in layered structures may possess diverse materials and geometry to realize specific application. For the hybridized structures, geometry and material properties relative to the middle surface are generally unsymmetrical, which have a significant impact on stability. Some models might lose capability to deal with such issues, so that these issues are less reported. Within the developed models, Reddy’s model possesses merit of simplicity and efficiency, so a Reddy-type higher-order zig-zag model is constructed by utilizing the proposed zig-zag function (ZZF). Instead of the standard finite element formulation using the principle of minimum potential energy, the three-field Hu–Washizu (HW) mixed variational principle is employed to acquire the finite element formulation which can meet the harmonious conditions of transverse shear stress at the interface of adjacent layers. By investigating buckling behaviors of hybridized structures, performance of the proposed finite element formulation is appraised by comparing with the results obtained from the three-dimensional (3D) model as well as other models. Effect of boundary conditions (BCs), material properties, and span-to-thickness ratio on the buckling loads is also studied in detail. Numerical results show that buckling loads of hybridized structures are significantly impacted by the chosen parameters. The results acquired from proposed model are in very good agreement with those obtained from the layerwise (LW) model and the 3D finite element results.

Energy-decaying and momentum-conserving schemes for transient simulations with mixed finite elements

We present the four-field (of displacements, velocities, stresses and strains) and the three-field (of displacements, velocities and stresses) mixed variational formulations for structural dynamics, which can be used to derive in an elegant way the energy-decaying and momentum-conserving (EDMC) time-stepping schemes for transient simulations with mixed solid and structural finite elements. These mixed variational formulations are in this work applied for the derivation of EDMC schemes for the geometrically exact, recently proposed, mixed-hybrid, shell finite elements (with excellent performance), which combine mixed interpolations of the Hu–Washizu type or the Hellinger–Reissner type with the assumed natural strain concept. To this end, the first-order and the second-order accurate EDMC schemes are designed. The superior properties of the considered mixed-hybrid shell finite elements, which are reflected in statics by excellent convergence properties, ability to take very large solution steps, and low-sensitivity to mesh distortion, are in this manner extended for (long-term) transient simulations. Numerical examples, which illustrate transient simulations with the EDMC schemes and mixed-hybrid shell finite elements, show (among other effects) that the underlying structure of the shell motion is numerically preserved even for very large time steps.

Mixed Kirchhoff stress–displacement–pressure formulations for incompressible hyperelasticity

The numerical approximation of hyperelasticity must address nonlinear constitutive laws, geometric nonlinearities associated with large strains and deformations, the imposition of the incompressibility of the solid, and the solution of large linear systems arising from the discretisation of 3D problems in complex geometries. We adapt the three-field formulation for nearly incompressible hyperelasticity introduced in Chavan et al. (2007) to the fully incompressible case. The mixed formulation is of Hu–Washizu type and it differs from other approaches in that we use the Kirchhoff stress, displacement, and pressure as principal unknowns. We also discuss the solvability of the linearised problem restricted to neo-Hookean materials, illustrating the interplay between the coupling blocks. We construct a family of mixed finite element schemes (with different polynomial degrees) for simplicial meshes and verify its error decay through computational tests. We also propose a new augmented Lagrangian preconditioner that improves convergence properties of iterative solvers. The numerical performance of the family of mixed methods is assessed with benchmark solutions, and the applicability of the formulation is further tested in a model of cardiac biomechanics using orthotropic strain energy densities. The proposed methods are advantageous in terms of physical fidelity (as the Kirchhoff stress can be approximated with arbitrary accuracy and no locking is observed) and convergence (the discretisation and the preconditioners are robust and computationally efficient, and they compare favourably at least with respect to classical displacement–pressure schemes).

A Gradient Stable Node-Based Smoothed Discrete Shear Gap Method for Analysis of Reissner–Mindlin Plates

In this paper, a gradient stable node-based smoothed discrete shear gap method (GS-DSG) using 3-node triangular elements is presented for Reissner–Mindlin plates in elastic-static, free vibration, and buckling analyses fields. By applying the smoothed Galerkin weak form, the discretized system equations are obtained. In order to carry out the smoothing operation and numerical integration, the smoothing domain associated with each node is defined. The modified smoothed strain with gradient information is derived from the Hu–Washizu three-field variational principle, resulting in the stabilization terms in the system equations. The stabilized discrete shear gap method is also applied to avoid transverse shear-locking problem. Several numerical examples are provided to illustrate the accuracy and effectiveness. The results demonstrate that the presented method is free of shear locking and can overcome the temporal instability issues, simultaneously obtaining excellent solutions.

Optimal parameter for the stabilised five-field extended Hu–Washizu formulation

We present a mixed finite element method for the elasticity problem. We expand the standard Hu–Washizu formulation to include a pressure unknown and its Lagrange multiplier. By doing so, we derive a five-field formulation. We apply a biorthogonal system that leads to an efficient numerical formulation. We address the coercivity problem by adding a stabilisation term with a parameter. We also present an analysis of the optimal choices of parameter approximation.

A nine nodes solid-shell finite element with enhanced pinching stress

In this paper we present a low-order solid-shell element formulation—having only displacement degrees of freedom (DOFs), i.e., without rotational DOFs. The element has an additional middle node, that allows efficient and accurate analyses of shell structures using elements at extremely high aspect ratio. The formulation is based on the Hu–Washizu variational principle leading to a novel enhancing strain and stress tensor that renders the computation particularly efficient, with improved in-plane and out-of-plane bending behavior (Poisson thickness locking). The middle-node is endowed with only one degree of freedom, in the thickness direction, allowing the assumption of a quadratic interpolation of the transverse displacement. Unlike solid-shell finite elements reported previously in the literature and formulated under the hypothesis of plane stress or with enhanced assumed strain parameter, the new solid-shell element here mentioned uses a complete three-dimensional constitutive law and gives an enhanced pinching stress, thanks to the middle-node. Moreover, to handle the various locking problems that usually arise on solid-shell formulation, the reduced integration technique is used as well as the assumed shear strain method. Finally to assess the effectiveness and performance of this new formulation, a set of popular benchmark problems, involving geometric non-linear analysis as well as elastic-plastic behavior has been investigated.

Experiments and Analysis on Stability of the Sandwich Structures with Soft Core

In this paper, first a complete buckling experiment of the sandwich beams with the foam core is carried out, which includes the manufacturing of specimens and their experimental verification. Second, a refined sinusoidal zig-zag theory (RSZT) is established, which can describe the zig-zag effect during the in-plane compression of sandwich beam and accommodate the transverse shear free surface boundary conditions. Based on the established model combined with Hu–Washizu variational principle, a two-node beam element has been developed to address the buckling problem of the sandwich beams. Thus, the established beam element is able to accommodate interlaminar continuous conditions of transverse shear stress. Several examples have been investigated to validate the accuracy of the established method. The comparative analysis of the results including experimental data, the results acquired from three-dimensional finite element (3D-FEM) and diverse models has been made. Comparative analysis shows that the accurate buckling loads can be acquired from the established model. Nevertheless, other models discarding the continuous conditions of transverse stresses among the adjacent layers largely overestimate the critical loads.