Hybrid Stress Element Research Articles

Study on hybrid stress element of three-dimensional arbitrary polyhedron

In three-dimensional problems, regular tetrahedron, pentahedron, and hexahedron elements are used in standard finite element software. In practice, however, the three-dimensional model is irregular. Traditional regular elements cannot accurately describe model structure. The field function integral accuracy of traditional displacement finite elements is low, and the number of elements considerably affects the integral results. As the number of elements increases, the computational efficiency decreases as well. Therefore, based on the hybrid stress element method, this paper proposes a new numerical calculation element for three-dimensional problems–a three-dimensional arbitrary polyhedron element. The corresponding calculation scheme is the 3-Dimensional Voronoi Cell Finite Element Method (3DVCFEM). This paper first verifies the effectiveness of the scheme. The stress field model with a singular solution is further simulated, and the change rule of the number of stress terms is obtained. On this basis, 3DVCFEM is introduced into calculating octree elements to solve complex classification problems and difficult calculation of octree elements. Moreover, the efficiency of the octree elements is investigated.

A New Hybrid Stress Element with Fluid

A New Hybrid Stress Element with Fluid

THEORY AND APPLICATIONS OF INCOMPATIBLE AND HYBRID ELEMENT METHOD——IN MEMORY OF PROF. WU CHANGCHUN

This memorial paper gives an overview of Prof. Wu Changchun's main contributions to computational mechanics, especially his pioneering work on the incompatible and hybrid element method and its applications. The general formula for the incompatible shape functions and the optimization design method for hybrid stress elements proposed by Prof. Wu are still important means to develop high performance finite elements and are cited by peers constantly.

多边形应力杂交单元的接触算法研究

In the case of small elements, it is difficult to obtain accurate stress fields and stress concentration with existing nonuniform models. An optimization algorithm for the theory of the PHSEM (polygonal hybrid stress element method) was proposed with the direct restriction method. The advantages of the PHSEM in the construction of stress functions and the division of integral regions make it more suitable for complex model boundaries and material boundaries and easier to realize-element meshing. The complete computing program was made according to the theoretical analysis. The results show that, the PHSEM can obtain macroscopic nonlinear mechanical responses, highly accurate stress fields and obvious stress concentration phenomena in the powder compaction process, which provides an effective means for the solution of the complex optimization problems.

Development of a 4‐node hybrid stress tetrahedral element using a node‐based smoothed finite element method

SummaryIn this paper, a new 4‐node hybrid stress element is proposed using a node‐based smoothing technique of tetrahedral mesh. The conditions for hybrid stress field required are summarized, and the field should be continuous for better performance of a constant‐strain tetrahedral element. Nodal stress is approximated by the node‐based smoothing technique, and the stress field is interpolated with standard shape functions. This stress field is linear within each element and continuous across elements. The stress field is expressed by nodal displacements and no additional variables. The element stiffness matrix is calculated using the Hellinger‐Reissner functional, which guarantees the strain field from displacement field to be equal to that from the stress field in a weak sense. The performance of the proposed element is verified by through several numerical examples.

ADVANCES IN SHAPE-FREE FINITE ELEMENT METHODS: A REVIEW

As the most important tool for simulation and computation, the finite element method has been widely applied in engineering and scientific problems. However, this powerful method still suffers from some inherent deficiencies, such as the sensitivity problem to mesh distortion and so on, which have not been completely solved yet. This paper systematacially introduces some research developments on new finite element methods, i.e., the shape-free finite element methods, including the hybrid stress-function elements for plane and 2D fracture problems, the hybrid displacement-function elements for Reissner-Mindlin plate bending problem, and new unsymmetric finite element for plane and 3D elasticity. Based on the existing hybrid stress and unsymmetric finite element methods, some advanced techniques, including the analytical trial function method, new natural coordinate method, generalized conforming technique, etc., were adopted in the aforementioned new approaches. The resulting new finite element models possess high precision and good robustness. In particular, they can keep their level of performance even in extremely distorted meshes. Furthermore, some historical challenges in the finite element method, such as the limitation defined by the MacNeal's theorem, the edge effect problem of the Resissner-Mindlin plate, were also successfully solved. At the end of this paper, some features of the new methods and further development are discussed.

A Refined Higher-Order Hybrid Stress Quadrilateral Element for Free Vibration and Buckling Analyses of Reissner-Mindlin Plates

This paper is devoted to develop a new 8-node higher-order hybrid stress element (QH8) for free vibration and buckling analysis based on the Mindlin/Reissner plate theory. In particular, a simple explicit expression of a refine method with an adjustable constant is introduced to improve the accuracy of the analysis. A combined mass matrix for natural frequency analysis and a combined geometric stiffness matrix for buckling analysis are obtained using the refined method. It is noted that numerical examples are presented to show the validity and efficiency of the present element for free vibration and buckling analysis of plates. Furthermore, satisfactory accuracy for thin and moderately thick plates is obtained and it is free from shear locking for thin plate analysis and can pass the nonzero shear stress patch test.

Higher-order hybrid stress triangular Mindlin plate element

A 6-node triangular hybrid stress element is presented for Mindlin plate in this paper. The proposed element, denoted by $$\hbox {TH6-27}\upbeta $$TH6-27β, can pass both the zero shear stress patch test and the non-zero constant shear stress enhanced patch test and, it can be employed to analyze very thin plate. To accomplish this purpose, special attention is devoted to selecting boundary displacement interpolation and stress approximation in domain. The arbitrary order Timoshenko beam function is used successfully to derive the displacement interpolation along each side of the element. According to the equilibrium equations, an appropriate stress approximation is rationally obtained. The assumed stress field is modified by using $$27\upbeta $$27β instead of $$15\upbeta $$15β to improve the accuracy. Numerical results show that the element is free of shear locking, and reliable for thick and thin plates. Moreover, it has no spurious zero energy modes and with geometric invariance (coordinate invariance, node sequencing independence).

An Adaptive Hybrid Stress Transition Quadrilateral Finite Element Method for Linear Elasticity

In this paper, we discuss an adaptive hybrid stress finite element method on quadrilateral meshes for linear elasticity problems. To deal with hanging nodes arising in the adaptive mesh refinement, we propose new transition types of hybrid stress quadrilateral elements with 5 to 7 nodes. In particular, we derive a priori error estimation for the 5node transition hybrid stress element to show that it is free from Poisson-locking, in the

St Venant Torsion and Bending of Prismatic Composite Shafts

This paper discusses the use of high-performance incompatible, enhanced-strain and hybrid stress elements for the St Venant torsion and bending of composite prismatic shafts, formulated in terms of the warping function. The properties of the shaft are uniform along its length but piece-wise constant and orthotropic across its cross-section. In order to analyse composite shafts whose microstructural details are difficult to be modelled directly by finite element discretisation, the two-scale asymptotic (or mathematical) homogenisation approach is also discussed; the resulting equilibrium equations can be readily solved by the aforementioned high-performance elements

Optimal fiber orientation and topology design for compliant mechanisms with fiber-reinforced composites

An approach for designing compliant mechanisms with glass fiber-reinforced epoxy materials is presented to obtain the optimum fiber orientation and topology structure simultaneously in this paper. Four-node hybrid stress elements and nodal design variables are adopted to suppress the islands and checkerboard phenomenon without additive filter technology and constraint. Taking fiber orientation and relative density as design variables, minimizing the weighted linear combination of the mutual strain energy and the strain energy is considered as objective function to achieve the desired deformation and enough load-carrying capacity of compliant mechanisms with the volume constraint. The displacement field of structure is obtained by the finite element analysis, and the non-linear optimization problem is solved via the well-known method of moving asymptotes. The numerical examples of designing compliant inverters and grippers with different weighted factors are investigated to demonstrate the effectiveness of the proposed method.

Field intensity factors around inclusion corners in 0–3 and 1–3 composites subjected to thermo-mechanical loads

A super inclusion corner apex element for polygonal inclusions in 0–3 and 1–3 composites is developed by using numerical stress and displacement field solutions based on an ad hoc finite element eigenanalysis method. Singular stresses near the apex of inclusion corner under thermo-mechanical loads can be obtained by using a super inclusion corner apex element in conjunction with hybrid-stress elements. The validity and the applicability of this technique are established by comparing the present numerical results with the existing solutions and the conventional finite element solutions. As examples of applications, a square array of square inclusions in 0–3 composites and a rectangular array of rectangular inclusions in 1–3 composites are considered. All numerical examples show that the present numerical method yields satisfactory solutions with fewer elements and is applicable to complex problems such as multiple singular points or fields in composite materials.

The Incremental Hybrid Natural Element Method for Elastoplasticity Problems

An incremental hybrid natural element method (HNEM) is proposed to solve the two-dimensional elasto-plastic problems in the paper. The corresponding formulae of this method are obtained by consolidating the hybrid stress element and the incremental Hellinger-Reissner variational principle into the NEM. Using this method, the stress and displacement variables at each node can be directly obtained after the stress and displacement interpolation functions are properly constructed. The numerical examples are given to show the advantages of the proposed algorithm of the HNEM, and the solutions for the elasto-plastic problems are better than those of the NEM. In addition, the performance of the proposed algorithm is better than the recover stress method using moving least square interpolation.

A systematic and quantitative method to determine the optimal assumed stress fields for hybrid stress finite elements

A novel method is developed to determine the optimal stress fields for the hybrid stress element. It provides a straightforward way as to how and why the resulting element can improve its displacement counterpart. A new inner product with material weighting matrix is defined to derive this quantitative method. It reveals the relationship in quantity of exact similarity degrees between different stress modes. It is different from the methods based on the conventional energy product which can only tell whether or not the stress and strain are orthogonal to each other because they are considered as mathematical vectors without any physical meaning. The strategy including two steps is proposed to determine the desired stress field. Firstly, the basic stress modes are broken into a set of sub-modes, where the necessary and unnecessary sub-modes are independent from each other because all of them are uniaxial. Secondly, all sub-modes are compared with their basic mode. The sub-mode with largest similarity degree with the basic mode implies that it represents the most important features inside the basic mode so it is selected as the optimal assumed stress mode for hybrid element. The 2D 4-node and 3D 8-node hybrid elements are illustrated by the present approach. Numerical examples are provided to compare the performances of element derived from different assumed stress fields.

24-DOF quadrilateral hybrid stress element for couple stress theory

A 24-DOF quadrilateral hybrid stress element for couple stress theory is proposed in this study. In order to satisfy the equilibrium equation in the domain of the element, the $$21\beta $$ 21 β Airy stress functions are chosen a assumed stress interpolation functions, and beam functions are adopted as the displacement interpolation functions on the boundary. This element can satisfy weak C $$^{0}$$ 0 continuity with second-order accuracy and weak $$\hbox {C}^1$$ C 1 continuity simultaneously. So the element can pass the enhanced patch test of a convergence condition. Moreover, the reduced integration and a stresses smooth technique are introduced to improve the element accuracy. Numerical examples presented show that the proposed model can pass the $$\hbox {C}^{0-1}$$ C 0 - 1 enhanced patch test and indeed possesses higher accuracy. Besides, it does not exhibit extra zero energy modes and can capture the scale effects of microstructure.

Refined 18-DOF triangular hybrid stress element for couple stress theory

Based on the couple stress theory and the hybrid stress element method proposed by Pian, a refined 18-DOF triangular hybrid stress element is established. The 15β Airy stress functions which satisfy the equilibrium equation in the domain of the element are chosen as the assumed stress interpolation functions, and boundary displacement interpolation functions are chosen as beam function. The characteristics of this element are possessing second-order accuracy and C0 and C1 continuity in the weak form, satisfying convergence condition for the enhanced patch test and does not exhibits extra zero energy modes. Moreover, a refined assumed stress element method is developed to improve the element accuracy. Numerical examples presented show that the proposed model indeed possesses higher accuracy and can pass the enhanced patch test, and can capture the scale effects of microstructure.

An effective numerical analysis of singular stress fields in dissimilar material wedges under thermo-mechanical loads

A super wedge apex element for dissimilar material wedges is developed by using numerical stress and displacement field solutions with an ad hoc finite element eigenanalysis method. Singular stresses near the apex of an arbitrary dissimilar material wedge under thermo-mechanical loads can be obtained by using a super wedge apex element in conjunction with hybrid-stress elements. The validity and applicability of this technique is established by comparing the obtained results with existing asymptotic solutions. All numerical examples show that the present numerical method yields satisfactory solutions with fewer elements and is applicable to complex problems such as multiple singular points or fields.

A quadrilateral membrane hybrid stress element with drilling degrees of freedom

A new kind of quadrilateral assumed stress hybrid membrane element with drilling degrees of freedom and a traction-free inclined side has been developed based on an extended Hellinger-Reissner principle which is established by expanding the essential terms of the assumed stress field as polynomials in the natural coordinates of the element. The homogeneous equilibrium equations are imposed in a variational sense through the internal displacements which are also expanded in the natural coordinates, while the tractionfree conditions along the inclined side are satisfied exactly. The use of such special element in the finite element solution is shown to be highly accurate when only a very coarse element mesh is used for plates with V-shaped rounded notches or inclined sides.

Singular stress analysis of an anisotropic elastic medium containing polygonal holes using a novel hybrid finite element method

A novel hybrid finite element method based on a numerical procedure is proposed to compute singular field near V-shaped notch corners in an anisotropic material containing polygonal holes. The finite element method is established by the following three steps: (1) an ad hoc one-dimensional finite element formulation is employed to determined numerical eigensolutions of the singular field near an V-shaped notch corner; (2) a super corner tip element is constructed to determine the strength of the singular field, in which the independent assumed stress fields are extracted from the eigensolutions; (3) a novel hybrid finite element equation is obtained by coupling the super corner tip element with the conventional hybrid stress elements. In numerical examples, generalized stress intensity factors for interactions between two polygonal holes with various geometry, space position and material property are mainly discussed. All the numerical results show that present method yields satisfactory singular stress field solutions with fewer elements. Compared with the conventional finite element methods and integral equation methods, the present method is more suitable for dealing with micromechanics of anisotropic materials.

8- and 12-node plane hybrid stress-function elements immune to severely distorted mesh containing elements with concave shapes

By simply improving the first version of hybrid stress element method proposed by Pian, several 8- and 12-node plane quadrilateral elements, which are immune to severely distorted mesh containing elements with concave shapes, are successfully developed. Firstly, instead of the stresses, the stress function ϕ is regarded as the functional variable and introduced into the complementary energy functional. Then, the fundamental analytical solutions (in global Cartesian coordinates) of ϕ are taken as the trial functions for 2D finite element models, and meanwhile, the corresponding unknown stress-function constants are introduced. Thus, the resulting stress fields must be more reasonable because both the equilibrium and the compatibility relations can be satisfied. Thirdly, by using the principle of minimum complementary energy, these unknown stress-function constants can be expressed in terms of the displacements along element boundaries, which can be interpolated directly by the element nodal displacements. Finally, the complementary energy functional can be rewritten in terms of element nodal displacement vector, and thus, the element stiffness matrix of such hybrid stress-function (HS-F) element is obtained. This technique establishes a universal frame for developing reasonable hybrid stress elements based on the principle of minimum complementary energy. And the first hybrid stress element proposed by Pian is just a special case within this frame. Following above procedure, two 8-node and two 12-node quadrilateral plane elements are constructed by employing different fundamental analytical solutions of Airy stress function. Numerical results show that, the 8-node and 12-node models can produce the exact solutions for pure bending and linear bending problems, respectively, even the element shape degenerates into triangle and concave quadrangle. Furthermore, these elements do not possess any spurious zero energy mode and rotational frame dependence.