Element Stiffness Matrix Research Articles

Strength calculation algorithm for shell structures based on a four-node discretization element

The article describes a method for calculating shell structures taking into account transverse shear deformations during elastoplastic deformation. The geometric relationships between the increments of deformations in an arbitrary layer of the shell and the increments of the components of the displacement vector, the components of the vector of inclination angles of the normal are obtained on the basis of the hypothesis of S.P. Tymoshenko. The slope of the normal and the increment of the slope are performed from the original position of the normal. The physical relationships between the increments of the stress tensor components and the increments of the components of the deformation tensor at the loading step were obtained on the basis of the hypothesis that they were proportional, based on the deformation theory of plasticity. In the finite element implementation of the developed algorithm, a four-node fragment of the middle surface of the shell structure with nodes located at the vertices of the quadrangular fragment was used. The increments of the displacement vector components, their first derivatives with respect to curvilinear coordinates, as well as the increments of the components of the normal rotation angle vector were selected as the nodal variable parameters of the four-node finite element at the loading step. The size of the stiffness matrix of the four-node finite element was 44x44. The developed algorithm was tested on test problems for calculating a cylindrical shell using a duralumin alloy with a deformation diagram in the form of a two-link broken line as a material.

Spatial refinement grillage model of box-girder bridges

The objective of this study was to overcome the shortcomings of the current analysis and calculation methods of the spatial bar system model, plane beam grillage model and solid element model in the design of the box-girder bridge. For example, the spatial bar system model lacks refined analysis of spatial effects, such as the shear lag effect, effective distribution width problem, and eccentric load coefficient problem. The similarity of the plane beam grillage model on the principle of equivalence cannot accurately reflect the shear stress distribution of the box-shaped composite beam and the local force on the top and bottom plates. The solid element and shell element models cannot directly calculate the internal force of the cross section. The calculation efficiency is low, and it is difficult to combine with the overall calculation. This paper proposes a spatial grid element analysis model suitable for fine analysis of box-girder bridges, which has the completeness and reliability of the analysis results and makes up for the shortcomings of the current box-girder bridge analysis methods. Firstly, the stiffness matrix of the spatial grid element is deduced, and the analysis program of the combined box girder based on the spatial grid element is written using MATLAB software. Then an actual calculation example is analyzed, and the difference between the calculation result of the spatial grid element and the result of the literature is compared and analyzed. The correctness and applicability of the spatial grid element model is verified.

A family of quadratic finite volume element schemes for anisotropic diffusion problems on triangular meshes

In this paper, we propose and analyze a family of quadratic finite volume element schemes for anisotropic diffusion problems on triangular meshes. The present work covers a previous one Zhou and Wu (2020) where only the case of scalar diffusion coefficients is considered. The novelty of this paper is that, by the introduction of some special parameters involving the diffusion tensor and the mesh geometry, each entry of the element stiffness matrix can be split as three parts that are easily computed and analyzed. Thanks to this representation, we manage to obtain a sufficient condition that ensures the existence, uniqueness and coercivity result of the finite volume element solution on triangular mesh. Moreover, this sufficient condition has a simple, analytic and computable expression that can be easily judged on any triangular meshes with arbitrary full diffusion tensors. In particular, when the diffusion tensor is an identity matrix, this condition reduces to a geometric one and only relies on the interior angles of each triangular element, which is consistent with the result in (Zhou and Wu, 2020). Moreover, based on this sufficient condition, we obtain a minimum angle condition for the case of general diffusion tensors, which is better than some existing ones. By the coercivity result, the optimal H1 error estimate is also obtained. Finally, the theoretical results are verified by some numerical experiments.

A locking-free P0 finite element method for linear elasticity equations on polytopal partitions

Abstract This article presents a $P_0$ finite element method for boundary value problems for linear elasticity equations. The new method makes use of piecewise constant approximating functions on the boundary of each polytopal element and is devised by simplifying and modifying the weak Galerkin finite element method based on $P_1/P_0$ approximations for the displacement. This new scheme includes a tangential stability term on top of the simplified weak Galerkin to ensure the necessary stability due to the rigid motion. The new method involves a small number of unknowns on each element, it is user friendly in computer implementation and the element stiffness matrix can be easily computed for general polytopal elements. The numerical method is of second-order accurate, locking-free in the nearly incompressible limit, and ease polytopal partitions in practical computation. Error estimates in $H^1$, $L^2$ and some negative norms are established for the corresponding numerical displacement. Numerical results are reported for several two-dimensional and three-dimensional test problems, including the classical benchmark Cook’s membrane problem in two dimensions as well as some three-dimensional problems involving shear-loaded phenomena. The numerical results show clearly the simplicity, stability, accuracy and efficiency of the new method.

Substructuring multi-resolution topology optimization with template

In the multi-resolution topology optimization (MTOP) method, by decoupling the finite element mesh and discretization of density field, the finite element analysis is carried out with a coarser mesh (i.e., super-elements), and the computational cost is thus greatly reduced in the process of topology optimization. However, the elemental stiffness matrix is calculated each iteration using the average density of super-elements, and this treatment is actually not only inaccurate but also leads to the checkerboard phenomenon and QR patterns when the filter radius is relatively small. In order to alleviate such issues, the super-element is treated as a substructure and the corresponding elemental stiffness matrix is obtained using static condensation. Furthermore, a template library is developed for the substructure based on its density distribution during the topology optimization process. By this means, the elemental stiffness matrix is not required to be calculated repeatedly, the accuracy of the MTOP method is improved significantly and the checkerboard patterns are effectively inhibited as well.

Static bending analysis of functionally graded sandwich beams using a novel mixed beam element based on first-order shear deformation theory

In this paper, the static bending behavior of the functionally graded sandwich beams is investigated using a novel mixed beam element based on the first-order shear deformation theory. The proposed beam element consists of two nodes and three degrees of freedom per node as the traditional Timoshenko beam element. By introducing a special process, selective or reduced integration are not required to calculate the element stiffness matrix and nodal force vector. The efficiency and accuracy of the proposed element are verified via some comparison studies. Then the proposed element is applied to study the bending behavior of the functionally graded sandwich beams with various boundary conditions. The influences of some parameters such as the power-law index, the slender ratio and the boundary conditions are studied carefully. The comparison studies and numerical results present that the new beam element is compatible with the static bending analysis of the functionally graded sandwich beams with a very coarse mesh. Besides, the numerical results show that the stress concentration may occur on the separated surface of the core layer and two face sheets, which should be noticed in practical applications of the FG sandwich beams.

Solid Truss to Shell Numerical Homogenization of Prefabricated Composite Slabs.

The need for quick and easy deflection calculations of various prefabricated slabs causes simplified procedures and numerical tools to be used more often. Modelling of full 3D finite element (FE) geometry of such plates is not only uneconomical but often requires the use of complex software and advanced numerical knowledge. Therefore, numerical homogenization is an excellent tool, which can be easily employed to simplify a model, especially when accurate modelling is not necessary. Homogenization allows for simplifying a computational model and replacing a complicated composite structure with a homogeneous plate. Here, a numerical homogenization method based on strain energy equivalence is derived. Based on the method proposed, the structure of the prefabricated concrete slabs reinforced with steel spatial trusses is homogenized to a single plate element with an effective stiffness. There is a complete equivalence between the full 3D FE model built with solid elements combined with truss structural elements and the simplified homogenized plate FE model. The method allows for the correct homogenization of any complex composite structures made of both solid and structural elements, without the need to perform advanced numerical analyses. The only requirement is a correctly formulated stiffness matrix of a representative volume element (RVE) and appropriate formulation of the transformation between kinematic constrains on the RVE boundary and generalized strains.

Complete vibration band gap characteristics of two-dimensional periodic grid structures

In this article, the spectral element method combined with Bloch theorem is applied to calculate complete vibration band gap characteristics of two-dimensional periodic grid structures, which is verified by structural vibration transmission results based on spectral element method and finite element method. According to the differential equation of motion, complete spectral stiffness matrix of beam element is established at first, then stiffness matrix of grid structure can be obtained by coordinate transformation and assembly method, so as to form governing equation of motion based on Bloch boundary conditions. The band gap characteristics of periodic structures can be calculated by solving motion equation. Through band gap distribution analysis of two-dimensional periodic grid structures, the dispersion relations of different unit arrangements and effects of material and structural parameters on band gap characteristics are obtained which can be applied to acoustic design or vibration and noise reduction areas.

Exact transcendental stiffness matrices of general beam-columns embedded in elastic mediums

Stiffness matrices of beams embedded in an elastic medium and subjected to axial forces are considered. Both the bending and the axial deformations have been incorporated. Two approaches for deriving the element stiffness matrix analytically have been proposed. The first approach is based on the direct force–displacement relationship, whereas the second approach exploits shape functions within the finite element framework. The displacement function within the beam is obtained from the solution of the governing differential equation with suitable boundary conditions. Both approaches result in identical expressions when the exact transcendental displacement functions are used. Exact closed-form expressions of the elements of the stiffness matrix have been derived for the bending and axial deformation. Depending on the nature of the axial force and stiffness of the elastic medium, seven different cases are proposed for the bending stiffness matrix. A unified approach to the non-dimensional representation of the stiffness matrix elements and system parameters that are consistent across all the cases has been developed. Through Taylor-series expansions of the stiffness matrix coefficients, it is shown that the classical stiffness matrices appear as an approximation when only the first few terms of the series are retained. Numerical results shown in the paper explicitly quantify the error in using the classical stiffness compared to the exact stiffness matrix derived in the paper. The expressions derived here gives the most comprehensive and consistent description of the stiffness coefficients, which can be directly used in the context of finite element analysis over a wide range of parameter values.

Improving convergence by optimizing the condition number of the stiffness matrices arising from least-squares finite element methods

The least-squares finite element method (LSFEM) has drawn much attention with desirable properties, such as always symmetric positive-definite stiffness matrix and approximation of primal and non-primal variables with equivalent accuracy. Despite a condition number comparable to that of Galerkin FEM, of O(h−2), it is sometimes found that LSFEM with low-order elements does not show a satisfactory accuracy of solution. In this report, we propose an easy-to-implement technique that optimizes the condition number of the stiffness matrix. The least-squares functional is applied to the first-order governing equations and a linear combination of the weak form domain and boundary equations, with normalization (i.e., weighting) parameters is constructed. The impact of these parameters on the condition number of the resulting global stiffness matrix is proved to be independent of mesh size or shape function order. By use of a global optimization scheme that tunes these parameters, we observe notable decrease in the condition number compared to cases where the least-squares principle is directly applied to the governing equations, and hence convergence is significantly improved. It is further shown that the technique can be applied to the elemental stiffness matrix for a computationally efficient means of determining normalization parameters. To show the general applicability of the method, we also apply this technique to solve for a two-dimensional (2D) lid-driven flow problem and to calculate the electromagnetic field distribution in a 2D rectangular resonant cavity. In both problems, the LSFEM solution accuracy is poor prior to optimization of weighting parameters, and accuracy does not increase much by mesh refinement alone. Moreover, the process can be generalized to Galerkin FEM and other methods to improve convergence properties.

Approximating inverse FEM matrices on non-uniform meshes with {mathcal{H}}-matrices

We consider the approximation of the inverse of the finite element stiffness matrix in the data sparse {mathcal{H}}-matrix format. For a large class of shape regular but possibly non-uniform meshes including algebraically graded meshes, we prove that the inverse of the stiffness matrix can be approximated in the {mathcal{H}}-matrix format at an exponential rate in the block rank. Since the storage complexity of the hierarchical matrix is logarithmic-linear and only grows linearly in the block-rank, we obtain an efficient approximation that can be used, e.g., as an approximate direct solver or preconditioner for iterative solvers.

Formation of Stiffness Matrices of Plane Elements Based on High-Order Degree Polynomials

In this paper, the method for forming the stiffness matrix of a plane element in a plane stress-strain state using functions approximating the displacements of the points of the element in the form of high-order degree polynomials is considered.

A Multi-State Strategy for Structural Damage Detection Using Sensitivity of Weighted Transmissibility Function

In practice, a model-based structural damage detection (SDD) method is helpful for locating and quantifying damages with the aid of reasonable finite element (FE) model. However, only limited information in single or two structural states is often used for model updating in existing studies, which is not reasonable enough to represent real structures. Meanwhile, as an output-only damage indicator, transmissibility function (TF) is proven to be effective for SDD, but it is not sensitive enough to change in structural parameters. Therefore, a multi-state strategy based on weighted TF (WTF) is proposed to improve sensitivity of TF to change in parameters and in order to further obtain a more reasonable FE model for SDD in this study. First, WTF is defined by TF weighted with element stiffness matrix, and relationships between WTFs and change in structural parameters are established based on sensitivity analysis. Then, a multi-state strategy is proposed to obtain multiple structural states, which is used to reasonably update the FE model and detect structural damages. Meanwhile, due to fabrication errors, a two-stage scheme is adopted to reduce the global and local discrepancy between the real structure and the FE model. Further, the [Formula: see text]-norm and the [Formula: see text]-norm regularization techniques are, respectively, introduced for both model updating and SDD problems by considering the characteristics of problems. Finally, the effectiveness of the proposed method is verified by a simply supported beam in numerical simulations and a six-storey frame in laboratory. From the simulation results, it can be seen that the sensitivity to structural damages can be improved by the definition of WTF. For the experimental studies, compared with the FE model updated from the single structural state, the FE model obtained by the multi-state strategy has an ability to more reasonably describe the change of states in the frame. Moreover, for the given structural damages, the proposed method can detect damage locations and degrees accurately, which shows the validity of the proposed method and the reliability of the updated FE model.

An unsymmetric 8‐node plane element immune to mesh distortion for linear isotropic hardening material

AbstractAn 8‐node quadrilateral plane element US‐QUAD8 is developed for linear isotropic hardening material in the framework of updated Lagrangian unsymmetric finite formulation where matrix is constructed by classical shape function, and matrix by the higher‐order Lagrangian basis function in global coordinate system. The present element eliminates the influence of Jacobian in the elemental stiffness matrix and guarantees the quadratic completeness of displacement field even under severe distorted mesh. Two typical problems meshed with angular/curved‐edge/mid‐side distorted elements demonstrate excellent performance of distortion resistance of the present element and good computational efficiency in severe mesh distortion.

The polygonal scaled boundary thin plate element based on the discrete Kirchhoff theory

The scaled boundary finite element method (SBFEM) is a powerful method for solving elastostatics problems based on polygonal elements. In this paper, we firstly construct the quadratic polygonal scaled boundary element only depends on the boundary nodal displacements by transforming the additional degrees of freedom derived from the constant body loads to those by the boundary nodes. Further, combining with the discrete Kirchhoff theory, we construct the polygonal scaled boundary thin plate element, which can possess the second order completeness. The element stiffness matrix for the thin plate problem can be transformed by the stiffness matrix for the plane problem directly by avoiding to compute the shape functions of SBFEM. Numerical examples verify that the proposed polygonal scaled boundary thin plate element has good accuracy.

Modeling multi-fracturing fibers in fiber networks using elastoplastic Timoshenko beam finite elements with embedded strong discontinuities — Formulation and staggered algorithm

To model fiber failures in random fiber networks, we have developed an elastoplastic Timoshenko beam finite element with embedded discontinuities. The method is based on the theory of strong discontinuities where the generalized displacement field is enhanced by a jump. The continuum mechanics formulation accounts for a fracture process zone and a bulk material while retaining traction continuity across the discontinuity. The additional degrees of freedom that are associated with the discontinuity are represented by a midpoint node, which is statically condensed to enable the implementation in commercial software through the user element interface. We propose a quasi-brittle fracture model, where the failure-related deformation is uncoupled from the plastic deformation in the bulk material. To retain the positive definite finite element stiffness matrix of the bulk material, we neglect the fracture-related softening of the discontinuity and employ a modified Newton iteration in the strain softening domain. Our implementation facilitates the integration into commercial finite element software and examples illustrate the robustness of the method. The FORTRAN source code is freely available to benchmark our model. We show that fiber failures contribute to the nonlinear stress–strain response of paper. Together with fiber–fiber bond failures, they can potentially explain the nonlinear stress–strain response of paper and nanopaper.

Structural health monitoring of splice joint in a steel beam

Nowadays, steel sections are extensively used in various constructions because of its lesser erection time and higher reliability. These members are connected either by welding or bolts. As time goes on, the structure degrades due to various reasons. In our study, the degradation of joints with bolted connections is presented. In reality, joints are damaged frequently when compared to the other parts of the structure. As bridges are generally of long spans, splicing is done to ensure the continuity of the spans and this splicing is done with the help of welding or bolts. Loosening of bolts is the most common damage that occurs in bolts, making the joint act like a semi-rigid joint. We have considered a bridge model consisting of two beams connected by splicing with bolts’ help, and this entire modelling is done in ABAQUS 6.13. Our intention to study the behaviour of the joint in both damaged and undamaged case. Loosening of bolts is considered as damage, and deflection of the joint is considered as the extent of the damage. We have performed a static analysis of the model. Theoretical validation is also done by developing an elemental stiffness matrix with semi-rigid connections using fixity factors.

A weak Galerkin finite element method based on [formula omitted] virtual element for Darcy flow on polytopal meshes

This paper presents a weak Galerkin finite element method based on H(div) virtual element for Darcy flow on polytopal meshes. Specifically, the pressure is approximated by constant functions on both the polytopal elements and the element sides, respectively. Their discrete weak gradients are calculated in the local H(div) virtual element spaces whose basis functions never need to be explicitly computed. The method works on very general polytopal meshes. After obtaining the numerical pressure, the numerical velocity that is locally mass-conservative and has continuous normal fluxes is easily obtained by a simple L2 projection. The error estimates for the numerical pressure and numerical velocity are proved. The assembly of element stiffness matrix is introduced. Numerical experiments in two and three dimensions are reported to illustrate the good performance of the method.

Isogeometric analysis with the Reissner-Mindlin shell for hull structural mechanical analysis

Isogeometric Analysis (IGA) is an important emerging numerical method that is being studied in various field. IGA is proposed to seamlessly connect the Computer-Aided Engineering (CAE) and the Computer-Aided Design (CAD) systems, so that the costly mesh generating stage in traditional Finite Element Method (FEM) can be skipped to improve the efficiency of design and analysis. In this paper, an IGA approach is introduced to the hull structural mechanical analysis. A Reissner-Mindlin shell element is constructed in the IGA framework, and the expression of the element stiffness matrix and load vector are derived in detail. Using the proposed method, a typical hull block structure is analyzed under both bending and torsion load. The maximum relative error of the displacement results is 1.6% compared to the reference results obtained from ANSYS 17.0, and the correctness of the proposed method is verified. It also proves that the proposed method can be used in engineering practice, and can be used to realize the integration of design and analysis.

The stress analysis of a shear wall with matrix displacement method using rectangular finite element

The aim in this study is to numerically present some characteristic features of the rectangular finite element using the matrix displacement method and to show the utility of this element in plane stress problems compared to the finite element method. The paper consisted of three parts. In the first part, all of the finite element formulation steps from choosing the convenient coordinate system to creating element stiffness matrix are presented respectively. In the second part of the study, a static finite element analysis of the shear wall is also made by ANSYS Mechanical APDL. In the final part, the results (displacements, strains and stresses) obtained from the previous parts are compared with each other by the help of tables and graphics. The results show that this method is effective and preferable for the stress analysis of shell structures. Further studies should be conducted in order to indicate the efficiency of the matrix displacement method for the solution of different types of plane stress problems using different finite elements.