Hellinger-Reissner Variational Principle Research Articles

A novel hybrid-stress method using an eight-node solid-shell element for nonlinear thermoelastic analysis of composite laminated thin-walled structures

Composite laminated thin-walled structures, widely used in high-speed aircrafts, undergo a complex thermal–mechanical coupling environment. Geometrical nonlinearities with a thermal effect bring significant challenge to finite element analysis of structures. In this paper, a novel hybrid-stress method based on the solid-shell element is proposed for nonlinear thermoelastic analysis. An eight-node solid-shell element (CSSH8) is developed based on the assumed natural strain method and hybrid-stress formulations to overcome various locking problems and achieve an effective 3D simulation for structures with a large span-thickness ratio. The Green–Lagrange displacement-strain relation is selected to take the geometrical nonlinearities into account. The modified generalized laminate constitutive model is extended to consider both the thermal expansion and temperature-dependent material properties. A temperature variation along the laminate thickness can also be assumed in the constitutive model. Nonlinear thermoelastic equilibrium equations are derived using the Hellinger–Reissner variational principle, in which five different coupling cases for thermal–mechanical loads can be fully involved. Numerical examples demonstrate that the proposed method with CSSH8 element is insensitive to various distorted meshes and numerically robust to pass the buckling point; meanwhile large step sizes can be achieved in the path-following nonlinear thermoelastic analysis.

A convex cone programming based implicit material point method

For conventional Material Point Method (MPM), both explicit-based and implicit-based MPM have shortcomings: explicit MPM has high requirements on time steps, and implicit MPM has high requirements on convergence. To circumvent these limitations, this paper innovatively proposes a convex cone programming-based implicit MPM (CP-MPM) algorithm, which ensures excellent convergence of solving complex problems involving large deformation, regardless of the chosen time step. In the proposed CP-MPM, the governing equations are initially transformed into a stationary point of a multivariable functional, leveraging the generalized Hellinger-Reissner (HR) variational principle. This stationary point problem is subsequently reformulated as a min-max convex cone optimization problem, with constraints originating from elastoplastic constitutive equations. In addition, a novel particle-based adhesive-frictional contact algorithm is proposed to effectively tackle the interaction between MPM domain and rigid bodies. The contact inequality between material points and rigid bodies is transformed into convex cone constraints, which rigorously prevent material point penetration and facilitate the imposition of irregular boundary conditions. Both elastic and elastoplastic problems involving contact under static or dynamic loading are ultimately represented as standard second-order cone programming (SOCP) problems, which is effectively solved by employing the Primal-Dual Interior Point (PDIP) method. The robustness, accuracy and convergence of the proposed method are validated through a series of elastic and elastoplastic benchmark problems. All results demonstrate the CP-MPM is a very promising method for implicitly solving complex practices.

Symplectic solutions of the plane annular sectors in micropolar elasticity

The symplectic approach is utilized to derive the solutions of the plane annular sector in micropolar elasticity. According to the Hellinger-Reissner variational principle, the Hamiltonian canonical equations are obtained for the plane problem in an annular sectorial region. Through the method of separation of variables, we obtain an eigenvalue equation of the Hamiltonian operator matrix which is a system of linear differential equations with variable coefficients. By the method of series solutions, its general solutions can be expressed as the linear combination of Bessel functions and power functions. The corresponding eigensolutions for three types of homogeneous boundary conditions are derived. According to the adjoint symplectic orthogonality of the eigensolutions, the solution of the micropolar plane annular sector is expressed as a series expansion of eigensolutions. Numerical analyses are presented to demonstrate the validity of the proposed method and confirm its ability to accurately capture the size effect.

A family of three-dimensional virtual elements for Hellinger-Reissner elasticity problems

We present a family of Virtual Element Methods for three-dimensional linear elasticity problems based on the Hellinger-Reissner variational principle. A convergence and stability analysis is developed. Moreover, using the hybridization technique and exploiting the information derived from this procedure, we show how to compute a better approximation for the displacement field. The numerical experiments confirm the theoretical predictions.

A stable implicit nodal integration-based particle finite element method (N-PFEM) for modelling saturated soil dynamics

In this study, we present a novel nodal integration-based particle finite element method (N-PFEM) designed for the dynamic analysis of saturated soils. Our approach incorporates the nodal integration technique into a generalised Hellinger-Reissner (HR) variational principle, creating an implicit PFEM formulation. To mitigate the volumetric locking issue in low-order elements, we employ a node-based strain smoothing technique. By discretising field variables at the centre of smoothing cells, we achieve nodal integration over cells, eliminating the need for sophisticated mapping operations after re-meshing in the PFEM. We express the discretised governing equations as a min-max optimisation problem, which is further reformulated as a standard second-order cone programming (SOCP) problem. Stresses, pore water pressure, and displacements are simultaneously determined using the advanced primal-dual interior point method. Consequently, our numerical model offers improved accuracy for stresses and pore water pressure compared to the displacement-based PFEM formulation. Numerical experiments demonstrate that the N-PFEM efficiently captures both transient and long-term hydro-mechanical behaviour of saturated soils with high accuracy, obviating the need for stabilisation or regularisation techniques commonly employed in other nodal integration-based PFEM approaches. This work holds significant implications for the development of robust and accurate numerical tools for studying saturated soil dynamics.

A C0 continuous mixed FE formulation for bending of laminated composite plates based on unified HSDT

AbstractThis study proposes a unified C0 continuous mixed finite element (MFE) formulation for the accurate and efficient prediction of stress components in laminated composite plates relying on Higher Order Shear Deformation Theory (HSDT). This unified form of the MFE accepts any convenient function for the representation of transverse shear deformation. The Hellinger–Reissner variational principle is employed for the derivation of MFE equations within a two‐field formulation involving stress resultants along with kinematical variables. Thus, the displacement and stress resultant fields are obtained directly from the global solution of the system of equations. In this manner, the in‐plane stress components are calculated over constitutive relations at the nodes without any need for error‐prone spatial derivatives. Furthermore, the independent interpolation of kinematic and stress resultant type variables allows the numerical solution to overcome the shear‐locking problem and ensure C0 continuity requirement. Numerical examples include convergence and comparison tests of predicted displacements and stress components under various boundary conditions and material configurations. Various test cases are considered for both the thin and thick plates subjected to sinusoidal and uniformly distributed loads. It is demonstrated that the proposed MFE formulation can capture stress components with high accuracy while being computationally efficient.

Stress‐hybrid virtual element method on quadrilateral meshes for compressible and nearly‐incompressible linear elasticity

AbstractIn this article, we propose a robust low‐order stabilization‐free virtual element method on quadrilateral meshes for linear elasticity that is based on the stress‐hybrid principle. We refer to this approach as the stress‐hybrid virtual element method (SH‐VEM). In this method, the Hellinger–Reissner variational principle is adopted, wherein both the equilibrium equations and the strain‐displacement relations are variationally enforced. We consider small‐strain deformations of linear elastic solids in the compressible and near‐incompressible regimes over quadrilateral (convex and nonconvex) meshes. Within an element, the displacement field is approximated as a linear combination of canonical shape functions that are virtual. The stress field, similar to the stress‐hybrid finite element method of Pian and Sumihara, is represented using a linear combination of symmetric tensor polynomials. A 5‐parameter expansion of the stress field is used in each element, with stress transformation equations applied on distorted quadrilaterals. In the variational statement of the strain‐displacement relations, the divergence theorem is invoked to express the stress coefficients in terms of the nodal displacements. This results in a formulation with solely the nodal displacements as unknowns. Numerical results are presented for several benchmark problems from linear elasticity. We show that SH‐VEM is free of volumetric and shear locking, and it converges optimally in the norm and energy seminorm of the displacement field, and in the norm of the hydrostatic stress.

Mesh reduction methods for thermoelasticity of laminated composite structures: Study on the B-spline based State Space Finite Element Method and Physics-Informed Neural Networks

Calculating the mechanical characteristics such as interlaminar stresses of composite laminated structures is a major challenge. To avoid the disadvantages of the traditional finite element method (FEM) of discretizing the whole structure too cumbersome, we implemented a method which only need to discretize on horizontal planes. The state space differential equation is derived from the Hellinger–Reissner variational principle, and both quadratic and cubic B-splines are used to interpolate the in-plane physical quantities, while the precise integration method is carried out along thickness to obtain accurate numerical results efficiently. The method is hence treated as the B-spline based State Space FEM (B-SSFE). To further reduce degree of freedoms, we implemented another method based on physics-informed neural networks (PINNs), which is completely meshless. Deep Learning is combined with differential equations carrying physical information, and neural networks are trained by constraining the loss function. We calculated different boundary conditions for a 3D composite laminated plate under thermal loading. Comparison is conduct among B-SSFE, PINNs and FEM. Numerical results demonstrate that in terms of computational accuracy, B-SSFE can be more accurate than FEM dealing with complex boundary conditions, and PINNs can reflects the stress fields well.

Nodal integration-based particle finite element method (N-PFEM) for poro-elastoplastic modelling of saturated soils under large deformation

This paper presents the nodal integration-based particle finite element method (N-PFEM) for poro-elastoplastic analysis of saturated soils subject to large deformation, utilising the generalised Hellinger-Reissner variational principle to reformulate the governing equations for saturated soil dynamics into a min–max optimisation problem. With finite element discretisation and nodal integration over cells, the problem is transformed into a standard second-order cone programming problem, efficiently resolved using the advanced primal–dual interior point method. The N-PFEM method has several advantages, including the use of linear triangular elements without volumetric locking issues, the avoidance of regularisation techniques, and the elimination of tedious variable mapping after remeshing. The numerical model is validated for large deformation analysis of saturated soils with a series of benchmarks against available analytical and numerical solutions, with a case study of the deformation of an embankment considering stone column reinforcement also carried out. This N-PFEM framework offers an effective and efficient simulation approach for the evolutionary behaviour of saturated soils with large deformation in complex geotechnical configurations of practical relevance.

A rotation-free Hellinger-Reissner meshfree thin plate formulation naturally accommodating essential boundary conditions

A rotation-free Hellinger-Reissner meshfree thin plate formulation is proposed to naturally accommodate the essential boundary conditions in a variationally consistent way. In this approach, the Galerkin weak form is established based upon the Hellinger-Reissner variational principle, where the bending moment is expressed as the second order smoothed gradients which inherently embed the integration constraint and fulfill the variational consistency condition. Owing to the Hellinger-Reissner variational principle, the essential boundary conditions naturally arise in the weak form. Accordingly, the enforcement of essential boundary conditions has a similar form as that of the Nitsche's method, i.e., both have standard consistent and stabilized terms. Compared with the Nitsche's method, the costly second and higher order derivatives of traditional meshfree shape functions are replaced by the fast-evaluated second order smoothed gradients and their derivatives. Meanwhile, the stabilized term in proposed method does not involve any artificial parameter and thus eliminates the stabilization parameter-dependent issue in the Nitsche's formulation. Several examples are presented to illustrate the convergence, accuracy and efficiency of the proposed rotation-free Hellinger-Reissner meshfree thin plate formulation.

A globally convergent smoothing Newton method for the second order cone complementarity approach of elastoplasticity problems

A new algorithm for solving small-deformation elastoplasticity problems as second order cone complementarity problems (SOCCPs) is presented. The SOCCP is derived from the Hellinger–Reissner variational principle, and then solved with a smoothing Newton method that is modified from the globally convergent Qi-Sun-Zhou method. The method is compared with two well-established methods: the return-mapping method and the conic programming (CP) approach with primal–dual interior-point method. Numerical results indicate that the proposed method shares common characteristics of CP class methods and offers better convergence efficiency in some cases. Moreover, it is shown that the proposed method can be imbedded in the framework of conventional finite element codes with only a few modifications, which makes it easy-to-implement and adaptive to large-scale parallel computing.

Stress Analysis of Laminated HSDT Beams Considering Bending Extension Coupling

This study demonstrates a mixed finite element formulation procedure for the bending and stress analyses of laminated composite beams. The finite element method is based on the Hellinger-Reissner variational principle, while the beam assumptions are based on the Higher Order Shear Deformation Theory (HSDT). Reddy’s shear function is employed for the beam theory where the beam is discretized by two-noded linear elements. The displacements and stress resultants are obtained directly at the nodes according to the proposed mixed formulation. The validation of current study is performed by comparison and convergence analyzes for various lamination cases under different boundary conditions.

Modelling the pre- and post-failure behaviour of faulted rock slopes based on the particle finite element method with a damage mechanics model

We develop a novel numerical model incorporating a damage mechanics formulation into the particle finite element method for analysing the pre- and post-failure behaviour of faulted rock slopes. In this computational framework, the stress-driven strain localisation and damage evolution are modelled based on an isotropic damage model; discontinuity structures like fault zones are represented as thin continuum layers with equivalent mechanical properties; the particle finite element method is used to solve and track the large deformation of rock masses. In the paper, we first present the mathematical formulation of the proposed model in the context of the Hellinger-Reissner variational principle. We conduct a thorough validation of our model for simulating the damage of brittle materials against well-documented experimental datasets of different failure scenarios. We then apply the model to simulate the deformation and failure phenomena of faulted rock slopes including both the pre-failure progressive damage and post-failure transient runout, demonstrating the strong capability of our model in physically capturing the initiation, evolution, and consequence of catastrophic rock slope failure across multiple temporal scales. In addition, our simulations can realistically reproduce the slope displacement time series with the interplay between rock damage and fault reactivation explored. The present work has important implications for understanding the physical mechanisms that drive the progressive destabilisation and catastrophic failure phenomena of rock slopes in nature.

A novel hybrid isogeometric element based on two-field Hellinger–Reissner principle to alleviate different types of locking

In the present work, novel hybrid elements are proposed to alleviate the locking anomaly in non-uniform rational B-spline-based isogeometric analysis (IGA) using a two-field Hellinger–Reissner variational principle. The proposed hybrid elements are derived by adopting the independent interpolation schemes for displacement and stress fields. The key highlight of the present study is the choice and evaluation of higher-order terms for the stress interpolation function to provide a locking-free solution. Furthermore, the present study demonstrates the efficacy of the proposed elements with the treatment of several two-dimensional linear-elastic benchmark problems alongside the conventional single-field IGA, Lagrangian-based finite element analysis (FEA), and hybrid FEA formulation. It is shown that the proposed class of hybrid elements performs effectively for analyzing the nearly incompressible problem domains that are severely affected by volumetric locking along with the thin plate and shell problems where the shear locking is dominant. A better coarse mesh accuracy of the proposed method in comparison with the conventional formulation is demonstrated through various numerical examples. Moreover, the formulation is not restricted to the locking-dominated problem domains but can also be implemented to solve the problems of general form without any special treatment. Thus, the proposed method is robust, most efficient, and highly effective against both shear and volumetric locking.

A three-dimensional particle finite element model for simulating soil flow with elastoplasticity

Soil flow is involved in many earth surface processes such as debris flows and landslides. It is a very challenging task to model this large deformational phenomenon because of the extreme change in material configurations and properties when soil flows. Most of the existing models require a two-dimensional (2D) simplification of actual systems, which are however three-dimensional (3D). To overcome this issue, we develop a novel 3D particle finite element method (PFEM) for direct simulation of complex soil flows in 3D space. Our PFEM model implemented in a fully implicit solution framework based on a generalised Hellinger–Reissner variational principle permits the use of a large time step without compromising the numerical stability. A mixed quadratic-linear element is used to avoid volumetric locking issues and ensure computational accuracy. The correctness and robustness of our 3D PFEM formulation for modelling large deformational soil flow problems are demonstrated by a series of benchmarks against analytical or independent numerical solutions. Our model can serve as an effective tool to support the assessment of catastrophic soil slope failures and subsequent runout behaviours.

A mixed constant-stress smoothed-strain element with a cubic bubble function for elastoplastic analysis using second-order cone programming

This paper proposes a novel optimization-based finite element method with a mixed constant-stress smoothed-strain element to solve elastoplastic problems. The proposed computational framework has three advantages: (1) in the proposed mixed constant-stress smoothed-strain element, the strain is smoothed on the edge-based smoothing domain, and the stress is assumed to be constant in each smoothing domain; (2) the cubic bubble shape function is introduced at the centroid of the element to enrich the displacement field. This makes the proposed approach accurate and free of volumetric locking for low-order elements; (3) the discrete elastoplastic boundary-value problem is formulated as a standard second-order cone problem based on the generalized Hellinger–Reissner variational principle, which is quickly and efficiently solved by the primal-dual interior-point algorithm. The proposed approach is validated on several classical problems of geotechnical engineering: the bearing capacity of footing and pipe under small deformation and slope stability. All the obtained results demonstrate that the proposed method adopting the novel mixed constant-stress smoothed-strain element offers competitively high computational accuracy and efficiency.

Hellinger–Reissner variational principle for a class of specified stress problems

Abstract Aiming at the problem of the specified stress condition in a partial region of the structure, the Hellinger–Reissner (H–R) variational principle is studied to provide a theoretical basis for the finite element numerical analysis. By introducing the unknown non-elastic strain as an additional unknown quantity to fulfill the specified stress condition, the elastic mechanics governing equations for the specified stress problem are given. Both stress and unknown non-elastic strain are taken as independent variables to establish the complementary energy principle and virtual work equation which are equivalent to the elastic mechanical control equation of the specified stress problem. Based on the conventional H–R variational principle, using displacement, stress and unknown non-elastic strain as independent variables, a H–R variational functional that satisfies the specified stress conditions is established by using Lagrange multiplier method. Also the variational functional with displacement, elastic strain and unknown non-elastic strain as independent variables is deduced by transforming the stress into the elastic strain. The corresponding finite formulae are derived based on an intra-element stress hybridization method. The H–R variational principle for the specified stress problem takes non-elastic strain as an independent variable, so that the stress explicitly appears in the equilibrium equation of the element and structure, which expands the application range and capabilities of the existing variational principle and finite element method. The correctness and accuracy of the theory and algorithm are verified by numerical examples.

Virtual Element Methods for three-dimensional Hellinger-Reissner elastostatic problems

Abstract This note aims at illustrating the application of the Virtual Element Method to elasticity problems in mixed form, following the Hellinger-Reissner variational principle. In order to highlight the potential and the flexibility of our approach, we focus on a three-dimensional low-order Virtual Element scheme, but similar considerations apply to two-dimensional and higher-order methods.

Explicit structure-preserving discretization of port-Hamiltonian systems with mixed boundary control

In this contribution, port-Hamiltonian systems with non-homogeneous mixed boundary conditions are discretized in a structure-preserving fashion by means of the Partitioned FEM. This means that the power balance and the port-Hamiltonian structure of the continuous equations is preserved at the discrete level. The general construction relies on a weak imposition of the boundary conditions by means of the Hellinger-Reissner variational principle, as recently proposed in [Thoma et al., 2021]. The case of linear hyperbolic wave-like systems, including the elastodynamic problem and the Maxwell equations in 3D, is then illustrated in detail. A numerical example is worked out on the case of the wave equation.

Mixed finite element formulation for bending of laminated beams using the refined zigzag theory

This study presents a mixed finite element formulation for the stress analysis of laminated composite beams based on the refined zigzag theory. The Hellinger–Reissner variational principle is employed to obtain the first variation of the functional that is expressed in terms of displacements and stress resultants. Due to C0 continuity requirements of the formulation, linear shape functions are adopted to discretize the straight beam domain with two-noded finite elements. The proposed formulation is shear locking free from nature since it introduces displacement and stress resultant terms as independent field variables. A monolithic solution of the global finite element equations is preferred, hence the stress resultants are directly obtained from the solution of these equations. The in-plane strain measures of the beam are obtained directly at the nodes over the compliance matrix and stress resultants by avoiding error-prone spatial derivatives. Following, transverse shear stresses are calculated from the equilibrium equations at the post-processing level. This simple but effective finite element formulation is first verified and tested for convergence behavior. The robustness of the approach is shown through some examples and its accuracy in predicting the displacement and stress components is revealed.