Global Stiffness Matrix Research Articles

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.

Dynamic stiffness formulation and vibration analysis of coupled conical-ribbed cylindrical-conical shell structure with general boundary condition

In this paper, the dynamic stiffness method (DSM) is presented to investigate the vibration characterizes of coupled conical-ribbed cylindrical-conical shell structure with general boundary condition. Rib stiffeners are evenly distributed in the inner side of the cylindrical shell. Firstly, the coupled structure is divided into several components (i.e., conical and cylindrical shells, annular plate) according to the geometric properties and coupling boundary conditions. The displacement fields of the components are derived from the governing equations exactly. Then, the dynamic stiffness matrix is formulated based on the relationship between the displacements and forces on the boundaries of the components. The global dynamic stiffness matrix of the coupled structure is assembled in a similar way as used in the finite element method (FEM). In addition, the experimental model of conical-ribbed cylindrical-conical shell structure is built and vibration test is conducted. The accuracy and stability of the present method are validated by comparing the present results with the results obtained by FEM and experimental test. Several numerical examples are conducted to understand the effects of geometric parameters on the dynamic behaviors of the coupled structure, which can offer positive guides for acoustic design of the combined shell structure.

The influence of the process-induced deformed state on straightness of prestressing strands after manufacture

Based on the study of elastic-plastic deformation, the expressions of longitudinal force Nt , bending Mb and torque Mt moments in wire-cross sections subject to the process parameters of rope torsion are obtained: wire tensioning efforts, untwisting and preformation.A technique for determining the technological internal force factors (TIFF) in the rope cross-sections: the longitudinal forces ; bending and torque moments. Being guided by the theorem on unloading and FEM, the external deformed state of the rope unloading from TIFF is studied. The expression of the rope unloading deformations vector as a function of the TIFF vector using the global stiffness matrix of 4×4 rope cross-section corresponding to the elastic deformation is obtained.Based on the bending deformations, an expression was χ and ζ obtained, the parameter h (h is the segment arrow at the chord length of 1 m), which, according to the regulatory standard, characterizes the degree of the rope straightness after manufacture. The connection between the parameters of torsion (tension, untwisting, and preformation) and the parameter h of straightness is established. It is determined that the unstraightness is directly affected only by the deviation from the uniformity of tension. The developed theory allows to determine the permissible deviation from the non-uniformity of tension with the permissible value of the parameter h of straightness. For example, a prestressing strand of a structure 1+6, having a diameter of 6.9 mm is considered to be of high quality according to the standard, if h ≤ 25mm. This value h corresponds to the deviation of the tension of Δκ = 0,8 one of the 6 wires of the layer. With an increase in deviation of tension, h increases almost rectilinearly to Δκ ≃3. With further increase, a loop is Δκ formed – the loss of stability.The obtained theory of the external deformed state of unloading provides further opportunity to study the internal deformed-stress state of the rope wires due to process-induced forces, which opens the prospect of even a deeper study of operational strength and durability of ropes.

Transient stability analysis of composite hydrogel structures based on a minimization-type variational formulation

We employ a canonical variational framework for the predictive characterization of structural instabilities that develop during the diffusion-driven transient swelling of hydrogels under geometrical constraints. The variational formulation of finite elasticity coupled with Fickian diffusion has a two-field minimization structure, wherein the deformation map and the fluid-volume flux are obtained as minimizers of a time-discrete potential involving internal and external energetic contributions. To analyze the structural stability of a certain equilibrium state of the gel satisfying the minimization principle, we apply the local stability criterion on the incremental potential, which is based on the idea that a stable equilibrium state has the lowest potential energy among all possible states within an infinitesimal neighborhood. Using this criterion in a finite-element context, it is understood that bifurcation-type structural instabilities are activated when the coupled global finite-element stiffness matrix loses its positive definiteness. This concept is then applied to determine the onset and nature of wrinkling instabilities occurring in a pair of representative film-substrate hydrogel systems. In particular, we analyze the dependencies of the critical buckling load and mode shape on the system geometry and material parameters.

Time history response of an elastic thin plate on a transversely isotropic multilayered medium due to vertical loadings

An efficient theoretical solution is proposed to investigate the time history response of an elastic circular thin plate bonded on multilayered transversely isotropic soils. Based on the analytic layer-element method, dynamic differential equation governing time dependent deflection of thin plate is solved with the global stiffness matrix equation of layered soils. In this way the compatible deformation at the plate-soil interface can be rigorously satisfied. After adopting the numerical inversion of Laplace-Hankel integral transform, the displacement and stress responses of the plate-soil system are eventually obtained in the time domain. The numerical results are validated by comparing with the finite element method (FEM). Parametric studies are also carried out to elucidate the influence of the external load form, plate rigidity, material anisotropy and soft stratum on the time history behavior of plate-soil system. It’s indicated that significantly influence has been bought by the shallowed buried weak layer, and as the buried depth decreases the elastic plate becomes more flexible.

Determination of Transverse Shear Stiffness of Sandwich Panels with a Corrugated Core by Numerical Homogenization.

Knowing the material properties of individual layers of the corrugated plate structures and the geometry of its cross-section, the effective material parameters of the equivalent plate can be calculated. This can be problematic, especially if the transverse shear stiffness is also necessary for the correct description of the equivalent plate performance. In this work, the method proposed by Biancolini is extended to include the possibility of determining, apart from the tensile and flexural stiffnesses, also the transverse shear stiffness of the homogenized corrugated board. The method is based on the strain energy equivalence between the full numerical 3D model of the corrugated board and its Reissner-Mindlin flat plate representation. Shell finite elements were used in this study to accurately reflect the geometry of the corrugated board. In the method presented here, the finite element method is only used to compose the initial global stiffness matrix, which is then condensed and directly used in the homogenization procedure. The stability of the proposed method was tested for different variants of the selected representative volume elements. The obtained results are consistent with other technique already presented in the literature.

Immersed‐interface finite element method based on a nonconformal Petrov–Galerkin formulation

AbstractImmersed interface methods are becoming increasingly necessary to solve problems of practical importance. To be considered as replacement for conventional, interface‐fitted, finite element method (FEM), an immersed‐interface finite element method (IIFEM) has to exhibit equivalent accuracy and robustness. The paper develops such an IIFEM based on nonconformal Petrov–Galerkin (ncPG) formulation for two‐dimensional linear elasticity. First, a bimaterial patch test (BPT) is developed which, when satisfied, guarantees consistency of an FEM formulation for problems with two materials in a single element. Then, nodal trial functions that satisfy the BPT are derived for triangular elements with immersed interface. These trial functions satisfy the Hadamard interface conditions for elements with immersed interface but are nonconformal on element edges with an immersed interface. Conventional conformal test functions are used in the Petrov–Galerkin formulation. The proposed ncPG‐IIFEM is applied to a benchmark multimaterial problem. The solution using ncPG‐IIFEM converges optimally and the error in displacement as well as stress are similar to interface‐fitted FEM. The ncPG‐IIFEM also leads to a well‐conditioned linear system with the condition number of the global stiffness matrix similar to interface‐fitted FEM. Overall, the proposed ncPG‐IIFEM is shown to be straightforward in application, optimal in convergence, and robust.

A tangent finite-volume direct averaging micromechanics framework for elastoplastic porous materials: Theory and validation

In this communication, we present a reformulated finite-volume direct averaging micromechanics (FVDAM) framework for periodic multiphase materials with elastic-plastic phases. The original elastoplastic version is formulated using the secant stiffness matrix approach, which requires a computationally intensive solution procedure of the governing differential equations. For the first time, the reformulation makes use of the tangent plasticity matrix approach that incorporates linearities to the unit cell boundary value problem. The tangential FVDAM theory is implemented using a quasi-Newton-Raphson strategy that quantifies errors in the evaluation of surface-averaged stresses due to the linearization, hence allowing large load increments. The tangential formulation is vigorously and fully assessed vis-à-vis the secant approach for porous unit cells on several aspects, including the convergence of the algorithmic implementation with mesh and step sizes, solution accuracy, and computation times. The reformulation simplifies the solution to the governing differential equations relative to its predecessor, albeit at the cost of more computational efforts for the reforming and reinverting of the global tangent stiffness matrix (or the so-called tangent operator). Additionally, advantages of the FVDAM theory relative to the classical finite-element homogenization are highlighted. The tangent FVDAM is employed to demonstrate the plasticity-triggered architectural effects in the response of periodic porous materials under different loading modes. The common mechanisms responsible for differences in the elastic-plastic response are attributed to the effective and hydrostatic stress alteration in the matrix phase. The current work paves the ground for the incorporation of FVDAM into readily-available commercial computational tools through the user material subroutines that are based on tangent stiffness approaches, which will produce a paradigm shift for FVDAM to solve the challenging multiscale structural problem.

High-order free vibration analysis of elastic plates with multiple cutouts

This paper aims to conduct the free vibration analysis of single-layer and laminated composite plates with multiple cutouts based on the scaled boundary finite element method (SBFEM) incorporated with the precise integration method (PIM) and the technique of the degree of freedom transform. It is applicable to a variety of shaped plates containing cordiform, circular and other complicated cutouts. The key equations are formulated built on the two-dimensional model, and the discretization is carried out in terms of only three translational displacement components as the basic unknowns. The high-order spectral elements are applied in the proposed technique to accurately simulate the arc boundaries of cutouts. Characterized by the important semi-analytical feature of the SBFEM, responses of the transverse free vibration are accurately explored. The derivation of governing equations strictly follows the 3D theory of elasticity without introducing any assumptions. The general solution of the SBFEM key equation is in the form of the matrix exponent, which is calculated by the PIM. As a type of highly precise approach, the PIM is introduced to create the global stiffness matrix from the analytical exponential matrix, which can make sure the eigensolutions with enough accuracy. Aided by the methodology of the kinetic energy formula, the global mass matrix of the perforated plate structure is constructed. According to the thin plates, the degree of freedom transform is employed to lower the dimensions of eigenvalue equation and thus improve the computational efficiency. Numerical examples of flexible square and circular plates with various cutouts reveal that excellent agreement is achieved between the natural frequencies predicted by the developed method and those from other methods and exact solutions. Furthermore, the effect of boundary conditions, skew angles, thickness-to-length ratios and sizes of cutouts on distributions of vibration frequencies is discussed.

On the dynamics of nano-frames

Size-dependent dynamic responses of small-size frames are modelled by stress-driven nonlocal elasticity and assessed by a consistent finite-element methodology. Starting from uncoupled axial and bending differential equations, the exact dynamic stiffness matrix of a two-node stress-driven nonlocal beam element is evaluated in a closed form. The relevant global dynamic stiffness matrix of an arbitrarily-shaped small-size frame, where every member is made of a single element, is built by a standard finite-element assembly procedure. The Wittrick–Williams algorithm is applied to calculate natural frequencies and modes. The developed methodology, exploiting the one conceived for straight beams in [International Journal of Engineering Science 115, 14–27 (2017)], is suitable for investigating free vibrations of small-size systems of current applicative interest in Nano-Engineering, such as carbon nanotube networks and polymer-metal micro-trusses.

The application of the Wittrick-Williams algorithm for free vibration analysis of cracked skeletal structures

In this study, the dynamic stiffness method is applied to a cracked beam that incorporates Rayleigh bar theory and Timoshenko beam theory, and the Wittrick-Williams algorithm is used to pinpoint natural frequencies. The influence of a crack on the two components of the Wittrick-Williams algorithm, namely, the number of negative leading diagonal terms in the upper triangular form of the global dynamic stiffness matrix at a trial frequency, and the sum of frequencies of each clamped-clamped element of the structure exceeded by the trial frequency, is explored. For the first component, the leading diagonal terms relevant to the additional degrees of freedom introduced by the crack should be considered. While for the second component, it is shown that a crack element itself does not contribute to the second component. The influence of a crack element on the second component only lies in that the crack alters the discretization thus more beam elements are involved. The same philosophy applies to the introduction of an affiliated mass or spring-mass system. To verify this statement, the free vibration of a cracked single-storey frame carrying a roving mass with rotary inertia is studied. The frequency results obtained using the dynamic stiffness method incorporating the Wittrick-Williams algorithm are compared with those exported from a finite element model in ANSYS. Some discussions about the results are made. The effects of roving translational inertia and roving rotary inertia are studied.

Vibration analysis of combined conical-cylindrical shells based on the dynamic stiffness method

An analytic dynamic stiffness method (DSM) is extended to analyze the free and forced vibration of combined conical-cylindrical shells with general boundary conditions in this paper. Flügge shell theory is utilized to formulate the motion equations of each shell component. Based on the exact general solutions of the motion equations, the dynamic stiffness matrices for each shell segment are established separately and the global dynamic stiffness matrix for the whole structure is established by assembling the dynamic stiffness matrices for each shell segment similarly as finite element method (FEM) did. Natural frequencies and forced responses of the combined shells are obtained based on the global dynamic stiffness matrix. Through comparing vibration results of DSM with ones from open literature and FEM, rapid convergence, good accuracy, and high efficiency of the DSM are demonstrated. In the numerical examples, the influences of boundary conditions, axial and circumferential mode numbers, and semi-vertex angles on the free vibration are studied. The effects of direction and location of external force and structural damping on the forced vibration are also discussed.

Thermo-mechanical behaviour of multi-layered media based on the Lord-Shulman model

The relaxation time model is introduced to analyse the thermo-mechanical problem for multi-layered media in this paper. The Fourier heat conduction equation is extended by introducing a relaxation time based on the Lord-Shulman model. For multi-layered media, the analytical layer-element method is employed to establish the global stiffness matrix equations by assembling the correlative layer-elements in the Laplace-Hankel domain. Then, solutions for displacement and temperature in the time domain are obtained by solving the global matrix equations and implementing a numerical inversion procedure. For numerical implementations, computational overflows are avoided by eliminations of positive exponentials in the stiffness matrix, and numerical instabilities in special cases due to singularity are removed by replacing singular layer-elements with well-conditioned ones. Comparisons with the existing solutions and the FEM analysis demonstrate the accuracy and efficiency of the proposed method. Finally, a series of numerical examples are presented to discuss the effects of transient load type, stratification, relaxation time and the surface heat transfer coefficient.

Analysis and research on solution method of metal layer stress in fiber metal laminates

As a new type composite material, fiber metal laminates (FMLs) have been widely applied in the aerospace field. To further develop and apply this new composite material, a stress prediction method for the metal layer in the laminates was examined. Based on the laminate theory, the stress characteristic of each layer material was analyzed under an external load. By studying the classical laminate theory, it was found that the calculation method of the global stiffness matrix fails to satisfy the actual situation of the laminates. The concept of the equivalent stiffness matrix was therefore introduced to modify the solution of stiffness matrix and accurately predict the metal layer stress. The equivalent stiffness matrix was then obtained using the sub-laminate stiffness theory and the energy method. The sub-laminate stiffness method was improved to apply to the orthotropic FMLs, and the energy method was expanded to apply to FMLs from a three-dimensional perspective. Taking 2/1 and 3/2 laminates of FMLs as examples, the metal layer stress was determined using the optical strain method and was predicted by employing the classical laminate theory along with the two correction methods. The effectiveness and advancement of the modified model were confirmed by comparative analysis.

A stability-enhanced peridynamic element to couple non-ordinary state-based peridynamics with finite element method for fracture analysis

The non-ordinary state-based peridynamics (NSPD) is a promising method for fracture analysis, and it can incorporate the constitutive relationship of classical continuum mechanics in peridynamics. However, the high computational cost is one of the main reasons limiting its usage. To improve computational efficiency of NSPD, a stability-enhanced peridynamic (PD) element is proposed to couple NSPD with finite element method (FEM), primarily for fracture analysis. Only the areas where cracks may initiate or propagate are solved with the proposed PD element, while the rest of the model is composed of conventional finite elements. The main feature of the proposed PD element is attributed to that the bond interaction does not have to be parallel to the bond direction in NSPD, and the stability of the element is enhanced by introducing the improved hourglass method into the element to restrain the instability due to the zero-energy mode. Based on the equation of motion of NSPD and the principle of virtual work of the PD element, the stiffness matrix of the PD element is derived and the coupled NSPD and FEM method with a global stiffness matrix is thus established. Meanwhile, a two-step interface correction method is developed to improve the accuracy at the interface of the two domains. Subsequently, to verify the proposed coupling method, the uniaxial tension of a strip plate, the propagation of longitudinal wave, the compact tension test of a steel plate, and the single edge notched tension test of an orthotropic lamina are simulated. With the proposed coupling method, the fracture process is accurately simulated, while the computing time is significantly reduced by up to more than 80% when compared with the one using NSPD alone. Moreover, with the constitutive relationship represented by the elastic tensor, the orthotropic materials can be easily modeled in the proposed PD element, and the stress-based criteria can be used with minor modifications. The proposed coupled NSPD and FEM method has high accuracy, computational efficiency, and convenience, and it has the potential to be a useful tool for quantitative analysis of complex engineering fracture problems.

Free Vibration and Buckling Analysis of Stiffened Sandwich Plates with Repeated Fold

In this paper, complex-shaped sandwich folded plate structures with and without stiffeners have been analyzed using first-order shear deformation theory. Finite element analysis using eight-noded isoparametric serendipity elements has been used. Regular plate elements have been applied to model the plates and stiffeners. Transformation matrices have been used to form the global stiffness matrix. Effect of different boundary conditions, fiber angles and core thicknesses has been discussed. Different stiffener arrangements have been analyzed to study their effects. Numerical results reveal that the fixed edge of folded plate increases the stiffness prominently. Also, proper positioning of stiffeners can significantly increase the stiffness and hence the natural frequency and the buckling load.

Finite element method algorithm for geotechnical applications based on Runge-Kutta scheme with automatic error control

This paper introduces a novel explicit algorithm to solve the finite element equation linking the nodal displacements of the elements with the external forces applied via means of non-linear global stiffness matrix. The proposed method solves the equation using Runge-Kutta scheme with automatic error control. The method allows any Runge-Kutta scheme, with the paper demonstrating the algorithm efficiency for Runge-Kutta schemes of second to fifth order of accuracy. The paper discusses the theoretical background, the implementation steps and validates the proposed algorithm. The numerical tests show that the proposed method is robust and stable. In comparison to the iterative implicit methods (e.g. Newton-Raphson method), the new algorithm overcomes the problem of occasional divergence. Furthermore, considering the computation time, the fifth-order accurate scheme proves to be competitive with the iterative method. It seems that the proposed algorithm could be a powerful alternative to the standard solution procedures for the cases with strong nonlinearity, where the typical algorithms may diverge.

Finite volume based asymptotic homogenization theory for periodic materials under anti-plane shear

A finite volume based approach is employed in the solution of unit cell problems at different orders of the asymptotic field expansion to construct a homogenization theory for anti-plane shear loading of unidirectional fiber-reinforced periodic structures. This new construction complements and further extends our recent contribution to asymptotic homogenization based on locally-exact elasticity unit cell solutions, He and Pindera (2020), to unit cells with multiple inclusions of arbitrary shapes. The present approach builds upon the previously developed finite-volume direct averaging micromechanics theory applicable under uniform strain fields, and extends it to account for strain gradients and non-vanishing microstructural scale relative to structural dimensions. The unit cell problems at different orders of the asymptotic field expansion are solved by satisfying local equilibrium equations in each subvolume of the discretized microstructure in a surface-averaged sense. This facilitates construction of local stiffness matrices at the subvolume level and subsequent assembly into the global stiffness matrix for the unit cell response under uniform and gradient strain fields. Comparison of the calculated microfluctuation functions and associated stress fields under uniform and gradient strain fields with those reported in the literature verifies the finite volume asymptotic solutions. The new theory's ability to accurately recover local fields is further illustrated through comparison with the direct numerical solution of a periodic structure with varying number of inclusions under gradient loading. The proposed homogenization approach is an efficient and accurate alternative to current numerical techniques for the analysis of periodic materials experiencing strain gradients regardless of microstructural scale, inclusion shape and number, demonstrated by analyzing arrays characterized by elliptical and square inclusions and multi-inclusion unit cells.

Numerical simulation of the stability of three-dimensional elastic composite structures based on the finite element method

A numerical method for solving the problem of the stability theory of linearly elastic bodies with small deformations in a general three-dimensional formulation is considered. The problems of this class are poorly studied in contrast to the two-dimensional problems of stability theory. At the same time, classical approaches do not allow one to take into account the effect on the structural stability of various three-dimensional effects: areas of compounds, zones of defects, etc. The study formulates a variation formulation of the problem of the three-dimensional stability theory. Based on the finite element method, a numerical statement is obtained in the form of a generalized eigenvalue problem with symmetric global stiffness matrix. The application of the proposed method is demonstrated by the example of calculating the stability of a composite plate under longitudinal compression. The simulation was carried out using the SMCM software package developed at the Department of Calculus Mathematics and Mathematical Physics of Bauman Moscow State Technical University.

Exact free vibration analysis for membrane assemblies with general classical boundary conditions

This paper proposes exact dynamic stiffness formulations for membranes and their assemblies under any arbitrary classical boundary conditions. First, by taking exact solutions in one direction satisfying all possible opposite edge supports, we can derive exact general solutions of the Helmholtz equation for membrane vibration. Then, generic force and displacement boundary conditions in the other direction are expressed in terms of the general solutions. Finally, the dynamic stiffness matrices of rectangular membrane elements are formulated, which can be assembled directly and allows applications of arbitrary boundary conditions. As an accurate and efficient modal solution technique, the Wittrick-Williams (WW) algorithm is applied onto the global dynamic stiffness matrix of the final structure. The most important issue of the WW algorithm, J0 count, has been resolved with the analytical expressions derived for all possible cases. The proposed dynamic stiffness method (DSM) is then applied to several examples including individual membranes and their assemblies. High accuracy and exactness of the proposed method within the whole frequency range is demonstrated by comparison with the finite element method. Besides, interesting findings have been observed on repeated eigenvalues with distinct mode shapes corresponding to certain aspect ratio and tension ratio, where physical and mathematical understanding has been provided.