Showing papers on "Direct stiffness method published in 2019"

A new Fragile Points Method (FPM) in computational mechanics, based on the concepts of Point Stiffnesses and Numerical Flux Corrections

Abstract: In this paper, a new method, named the Fragile Points Method (FPM), is developed for computer modeling in engineering and sciences. In the FPM, simple, local, polynomial, discontinuous and Point-based trial and test functions are proposed based on randomly scattered points in the problem domain. The local discontinuous polynomial trial and test functions are postulated by using the Generalized Finite Difference method. These functions are only piece-wise continuous over the global domain. By implementing the Point-based trial and test functions into the Galerkin weak form, we define the concept of Point Stiffnesses as the contribution of each Point in the problem domain to the global stiffness matrix. However, due to the discontinuity of trial and test functions in the domain, directly using the Galerkin weak form leads to inconsistency. To resolve this, Numerical Flux Corrections, which are frequently used in Discontinuous Galerkin methods are further employed in the FPM. The resulting global stiffness matrix is symmetric and sparse, which is advantageous for large-scale engineering computations. Several numerical examples of 1D and 2D Poisson equations are given in this paper to demonstrate the high accuracy, robustness and convergence of the FPM. Because of the locality and discontinuity of the Point-based trial and test functions, this method can be easily extended to model extreme problems in mechanics, such as fragility, rupture, fracture, damage, and fragmentation. These extreme problems will be discussed in our future studies.

Prediction of the dynamic equivalent stiffness for a rubber bushing using the finite element method and empirical modeling

Abstract: A hybrid method using an approximation that is based on the finite element analysis and empirical modeling is proposed to analyze the dynamic characteristics of a rubber bushing. The hyperelastic–viscoplastic model and an overlay method are used to obtain the hysteresis of the rubber bushing in the finite element analysis. A spring, fractional derivatives, and frictional components are used in the empirical model to obtain the dynamic stiffness in wide ranges of the excitation frequencies and amplitudes. The parameters of the proposed empirical model are determined using the hysteresis curves that were obtained from the finite element analysis. The dynamic stiffness of the rubber bushing in the wide ranges of the frequencies and amplitudes was predicted using the proposed hybrid method and was validated using lower arm bushing experiments. The proposed hybrid method can predict the dynamic stiffness of a rubber bushing without the performance of iterative experiments and the incurrence of a high computational cost, making it applicable to analyses of full-size vehicles with numerous rubber bushings under various vibrational loading conditions.

A numerical formulation and algorithm for limit and shakedown analysis of large-scale elastoplastic structures

Abstract: In this paper, a novel direct method called the stress compensation method (SCM) is proposed for limit and shakedown analysis of large-scale elastoplastic structures. Without needing to solve the specific mathematical programming problem, the SCM is a two-level iterative procedure based on a sequence of linear elastic finite element solutions where the global stiffness matrix is decomposed only once. In the inner loop, the static admissible residual stress field for shakedown analysis is constructed. In the outer loop, a series of decreasing load multipliers are updated to approach to the shakedown limit multiplier by using an efficient and robust iteration control technique, where the static shakedown theorem is adopted. Three numerical examples up to about 140,000 finite element nodes confirm the applicability and efficiency of this method for two-dimensional and three-dimensional elastoplastic structures, with detailed discussions on the convergence and the accuracy of the proposed algorithm.

Higher order semi-analytical solution for bending of angle-ply composite laminated cylindrical shells based on three-dimensional theory of elasticity

Abstract: This paper develops a high-performance semi-analytical numerical model to analyze the bending responses of the angle-ply composite laminated cylindrical shells with the fiber reinforced layers using the scaled boundary finite element method (SBFEM). As the thin-walled structures, the angle-ply composite laminated shells are assumed to be made of orthotropic materials in the cylindrical coordinate system. Both the geometric and basic variables are discretized by utilizing the two-dimensional (2D) high order spectral elements in the curved surface domain of the shells. According to the exact three-dimensional (3D) theory of elasticity rather than the approximate shell theories, the weak form of the partial differential governing equations for each layer of the composite laminated cylindrical shells in the cylindrical coordinate system are transformed into ordinary differential equations using the SBFEM. In the circumstances, there are no variables about the curved surface of shell in the SBFEM governing equations for each lamina, so that it can be analytically solved on the basis of the dual variable approach and the precise integration technique (PIT). Employing the interface continuity conditions of displacement between the layers, the complete SBFEM model with respect to the global stiffness matrix of the composite laminated cylindrical shell can be obtained. Unlike the general layerwise theories, which supposes that the basic variables varied linearly with the thickness coordinate, the through-thickness distributions of the displacement field in each discrete layer is assumed to be a quadratic polynomial with respect to the radial coordinate in this paper, thus the through-thickness stress field can be described more accurate. Numerical examples for solving the bending problem of composite laminated cylindrical shells are presented. As a result, the numerical efficiency, accuracy and applicability of the proposed formulations are confirmed by the comparison of the published results involving the distributions of the displacements and stresses through the thickness and along the circumferential direction for different staking configurations, geometric properties and boundary conditions.

Coupled electromagnetic-thermal-fluidic analysis of permanent magnet synchronous machines with a modified model

Abstract: The researches on the heat generation and dissipation of the permanent magnet synchronous machines (PMSMs) are integrated problems involving multidisciplinary studies of electromagnetism, thermomechanics, and computational fluid dynamics. The governing equations of the multi-physical problems are coupled and hard to be solved and illustrated. The high accuracy mathematical model in the algebraically integral conservative forms of the coupled fields is established and computed in this paper. And the equation coupling with the fluid flow and the temperature variation is modified to improve the positive definiteness and the symmetry of the global stiffness matrix. The computational burden is thus reduced by the model modification. A 20kW 4500rpm permanent magnet synchronous machine (PMSM) is taken as the prototype, and the calculation results are validated by experimental ones.

Analysis and Modelling of Non-Fourier Heat Behavior Using the Wavelet Finite Element Method.

Abstract: Non-Fourier heat behavior is an important issue for film material. The phenomenon is usually observed in some laser induced thermal responses. In this paper, the non-Fourier heat conduction problems with temperature and thermal flux relaxations are investigated based on the wavelet finite element method and solved by the central difference scheme for one- and two-dimensional media. The Cattaneo–Vernotte model and the Dual-Phase-Lagging model are used for finite element formulation, and a new wavelet finite element solving formulation is proposed to address the memory requirement problem. Compared with the current methodologies for the Cattaneo–Vernotte model and the Dual-Phase-Lagging model, the present model is a direct one which describe the thermal behavior by one equation about temperature. Compared with the wavelet method proposed by Xiang et al., the developed method can be used for arbitrary shapes. In order to address the efficient computation problems for the Dual-Phase-Lagging model, a novel iteration updating methodology is also proposed. The proposed iteration algorithms on time avoids the use the global stiffness matrix, which allows the efficient calculation for title issue. Numerical calculations have been conducted in the manner of comparisons with the classical finite element method and spectral finite element method. The comparisons from accuracy, efficiency, flexibility, and applicability validate the developed method to be an effective and alternative tool for material thermal analysis.

Rapid non-linear finite element analysis of continuous and discontinuous Galerkin methods in MATLAB.

Abstract: MATLAB is adept at the development of concise Finite Element (FE) routines, however it is commonly perceived to be too inefficient for high fidelity analysis. This paper aims to challenge this preconception by presenting two optimised FE codes for both continuous Galerkin (CG) and discontinuous Galerkin (DG) methods. Although this has previously been achieved for linear-elastic problems, no such optimised MATLAB script currently exists, which includes the effects of material non-linearity. To incorporate these elasto-plastic effects, the externally applied load is split into a discrete number of loadsteps. Equilibrium is determined at each loadstep between the externally applied load and the arising internal forces using the Newton–Raphson method. The optimisation of the scripts is primarily achieved using vectorised blocking algorithms, which minimise RAM-to-cache overheads and maximise cache reuse. The optimised codes yielded maximum speed gains of × 25.7 and × 10.1 when compared to the corresponding unoptimised scripts, for CG and DG respectively. It was identified that with increasing refinement of the mesh, the solver time begins to dominate the overall simulation time. This bottleneck has a greater disadvantage on the DG code, predominantly due the asymmetric nature of the global stiffness matrix. The implementation of an efficient solver would see further improvement to the overall run times, particularly for large problems.

Advancement of MSA-Technique for Stiffness Modeling of Serial and Parallel Robotic Manipulators

Abstract: The paper presents advancement of the matrix structural analysis technique (MSA) for stiffness modeling of robotic manipulators. In contrast to the classical MSA, it can be applied to both parallel and serial manipulators composed of flexible and rigid links connected by rigid, passive or elastic joints with multiple external loadings. The manipulator stiffness model is presented as a set of basic equations describing the link elasticities that are supplemented by a set of constraints describing connections between links. These equations are aggregated straightforwardly in a common linear system without traditional merging of the matrix rows and columns, which allows avoiding conventional manual transformations at the expense of numerical inversion of the sparse matrix of higher dimension.

Method for detecting structural damage of bridge by using test vehicle

Abstract: The invention belongs to the technical field of bridge structure diagnosis and relates to a method for detecting the structural damage of a bridge by using a test vehicle. The method includes the following steps that: (1) based on a vehicle-bridge coupling system, the vertical acceleration of a stationary test vehicle is obtained on the basis of external excitation change; (2) the vertical acceleration response of the contact point of the test vehicle and the surface of the bridge under a condition that the test vehicle is stationary is indirectly calculated through the vertical acceleration of the stationary test vehicle; (3) a transfer rate function matrix is established for the obtained vertical acceleration response of the contact point through a transfer rate function-based method; (4) the nth-order frequency and mode of the bridge are solved based on the transfer rate function matrix; and (5) stiffness inversion is performed on the nth-order frequency and mode of the bridge through an improved direct stiffness method, so that the section bending stiffness of each unit node can be identified. With the method of the invention adopted, shortcomings such as time-consuming and labor-consuming performance and relatively high cost of traditional manual detection can be eliminated, the bending stiffness of the bridge can be effectively identified, and the deflection deformation of the bridge under any load can be identified.

Exact Solutions for Buckling and Second-Order Effect of Shear Deformable Timoshenko Beam–Columns Based on Matrix Structural Analysis

Abstract: Shear deformable beams have been widely used in engineering applications. Based on the matrix structural analysis (MSA), this paper presents a method for the buckling and second-order solutions of shear deformable beams, which allows the use of one element per member for the exact solution. To develop the second-order MSA method, this paper develops the element stability stiffness matrix of axial-loaded Timoshenko beam–columns, which relates the element-end deformations (translation and rotation angle) and corresponding forces (shear force and bending moment). First, an equilibrium analysis of an axial-loaded Timoshenko beam–column is conducted, and the element flexural deformations and forces are solved exactly from the governing differential equation. The element stability stiffness matrix is derived with a focus on the element-end deformations and the corresponding forces. Then, a matrix structural analysis approach for the elastic buckling analysis of Timoshenko beam–columns is established and demonstrated using classical application examples. Discussions on the errors of a previous simplified expression of the stability stiffness matrix is presented by comparing with the derived exact expression. In addition, the asymptotic behavior of the stability stiffness matrix to the first-order stiffness matrix is noted.

Soil-structure interaction effects on the seismic performance of frame structures

Abstract: In this work, a numerical attempt has been elaborated to study the effects of soil properties, seismic impact and interaction between frame structure and soil foundation. The frame, soil and interface continuum have been modeled by using the finite element concept. In this approach, a numerical program has been established based on the direct method to analyze the soil-structure interaction under seismic loading. Adding, the developed thinlayer interface element ensuring the incompatibility of degrees of freedom between frame and soil elements is incorporated. Obtained results lead to explain (1) the influence of the soil-structure interaction on the seismic response of the frame structure and interface media, (2) the time history deflection of the superstructure and interface medium under seismic loading, (3) the effect of the interaction factor of soil-structure interaction on the seismic behavior of the interface medium and frame structure and (4) the earthquake intensity on the seismic damage. Lateral displacements at the top of the frame and at the interface level, and shear stresses at the contact between the foundation and structure are plotted and commented.

A general criterion for solid instability and its application to creases

Abstract: A general force-perturbation-based criterion for solid instability is proposed, which can predict instability including crease without priori knowledge of instability configuration. The crease instability is analyzed in detail, we found that the occurrence of solid instability does not always correspond to the non-positive definiteness of global stiffness matrix. An element stiffness-based criterion based on material stiffness is proposed as a stronger criterion in order to fast determine the occurrence of instability. This criterion has been shown to degenerate into the criterion for judging instability of certain known phenomena, such as necking and shear band phenomena. Besides, instability in strongly anisotropic materials is also predicted by the element stiffness-based criterion.

Fully coupled multi-physics nonlinear analysis of structural space frames subjected to fire using the direct stiffness method

Abstract: This article develops a fully coupled hydro-thermo-mechanical formulation based on the direct stiffness method for analysis of steel and reinforced concrete structural space frames. The superiority...

Application of a New Infinite Element Method for Free Vibration Analysis of Thin Plate with Complicated Shapes

Abstract: A novel infinite element method (IEM) is presented in this paper for solving plate vibration problems. In the proposed IEM, the substructure domain is partitioned into multiple layers of geometrically similar finite elements which use only the data of the boundary nodes. A convergence criterion based on the trace of the mass matrix is ​​used to determine the number of layers in the IE model partitioning process. Furthermore, in implementing the Craig-Bampton (CB) reduction method, the inversion of the global stiffness matrix is calculated using only the stiffness matrix of the first element layer. The validity and performance of the proposed method are investigated by means of four illustrative problems. The first example considers the case of a simple clamped rectangular plate. It is observed that the IEM results are in good agreement with the theoretical results for all six natural frequencies. The second example considers the frequency response of a clamped rectangular plate with a crack. The main feature of IEM is that a very fine and good quality virtual mesh can be created around the crack tip. The third and fourth examples consider the natural frequency of a multiple point supported plate and a perforated plate, respectively. The results are obtained just need to adjust the reference point or boundary nodes. The parametric analyses for various geometric profiles are easy to be conducted using these numerical techniques. In general, the results presented in this study confirm that the proposed IEM algorithm provides a fast, direct and accurate means of simulating the dynamic response of various plate structures.

Motor thermoanalysis method with temperature field directly coupled with heat circuit

Abstract: A motor thermal analysis method with a temperature field directly coupled with a thermal circuit is used for modeling partial parts of the motor, and a thermal circuit method is used for modeling other parts. A temperature field is contacted with a thermal circuit through an equivalent temperature boundary and an equivalent convection boundary. The thermal circuit part is composed of one-dimensional finite elements, and two connecting boundaries are deemed as two boundary elements. An element stiffness matrix, an element loading matrix and an element mass matrix corresponding to the one-dimensional finite elements and the boundary elements are respectively overlaid to a global stiffness matrix, a global loading matrix and a global mass matrix, and the distribution of temperature in the temperature field and the distribution of temperature in the thermal circuit are obtained simultaneously by solving a whole system of linear equations.

Numerical Solution of the Tri-harmonic K irchhoff Plate Equation Resulting from a Strain Gradient Theory

Abstract: A second gradient continuum theory is formulated based on second gradients of displacements. For a reduction of additional material parameters, the modified strain gradient model is used and a partial dierential equation of rank six is developed using the Kirchhoff plate assumptions. The solutions of the governing tri-harmonic plate bending equation incoorperate size-effects. Balance equations are presented and higher-order stress-strain relations are derived. In order to account for second gradients of displacements, which manifest themselves in the higher-order terms of a strain energy density, a C1–continuous displacement field is preferable. So-called Hermite finite element formulations allow for merging gradients between elements and are used to achieve global C1–continuity of the solution. Element stiffness matrices as well as the global stiffness matrix are developed for a lexicographical order of nodes and for equidistantly distributed elements. The convergence, the C1–continuity, and the size effect are demonstrated.

A finite element algorithm for solution of the natural vibration problem of electroelastic bodies with passive external electric circuits, interacting with a quiescent fluid

Abstract: In this paper we consider a mathematical statement of the problem on natural vibrations of piecewise-homogeneous electroelastic bodies with passive external electric circuits (shunting circuits) of arbitrary configuration and interacting with a quiescent fluid. The behavior of the piezoelectric body is described using the equations of electrodynamics of deformable electroelastic media in the quasi-static approximation. The motion of an ideal fluid in the case of small perturbations is considered within the framework of the acoustic approximation. Small strains in a thin plate are determined using the Reissner ? Mindlin theory. The numerical solution is developed using the finite element method. The proposed algorithm is based on the approach, in which the global stiffness matrix generated with the aid of the ANSYS software package is decomposed into required constituents. The system of governing equations is constructed using the developed algorithm, which is realized in the FORTRAN language. Complex eigenvalues of the examined system are defined from the solution of the non-classic modal problem using the Mueller method. A thin plate with piezoelectric element located on the free surface of a layer of a quiescent fluid of finite size is considered as an example.

A Novel Composit Vascularized FEM Model for Liver Surgery Simulation and Guidance

Abstract: In order to acquire comprehensive information from the limited visual feedback of liver surgery under endoscope, a liver model with high accuracy and low time consumption is indispensable. Many researchers has studied the biomechanical models of liver focusing on parenchyma elasticity and viscosity property in order to simulation the realtime deformation during the surgery, but only a few of them consider the vascular network. Some vascularized liver models contain ill-conditioned matrix issue that makes the choices of solver strictly limited. In the paper we propose a novel method to construct a real-time composite model in order to reduce this limitation. Instead of constructing the global stiffness matrix, we build FEM models for parenchyma and vessels separately and couple them with propagation of force and deformation. We apply a new mapping method to distribute the torque applied on the beam node to linear forces of the tetrahedron nodes. The simulation results show the feasibility of our method under certain precision.

Static and Dynamic Analysis of Flexure-based Compliant Mechanism by Matrix Displacement Method

Abstract: This paper proposed a novel analytical modeling method for the static and dynamic analysis for the flexure-based compliant mechanisms based on the matrix displacement method. Firstly, the theoretical compliance characteristics of a flexure element is transformed into its elemental stiffness matrix. Then, based on the elemental stiffness matrix, the new elemental stiffness matrix for two connected rigid bodies is derived by considering two points on rigid bodies and the forces applied on them as the nodal displacement and nodal forces. Finally, by expanding each new elemental stiffness matrix into contribution matrix and superimposing all of the contribution matrices, the global stiffness matrix can be obtained. The comparisons between the analytical method and the finite element analysis for two specific compliant mechanisms are conducted. The maximum differences of the analytical results with respect to the finite element analysis results are less than 5%, which demonstrate the high accuracy and effectiveness of the analytical model..

Application of Craig-Bampton Reduction Technique and 2D Dynamic Infinite Element Modeling Approach to Membrane Vibration Problems

Abstract: An approach is presented for solving membrane vibration problems using an integrated scheme consisting of the Craig-Bampton (CB) reduction technique and a 2D dynamic infinite element modeling (DIEM) method. In the proposed CB-DIEM scheme, the substructure domain is partitioned into multiple layers of geometrically-similar infinite elements (IEs) which use only the data of the boundary nodes. A convergence criterion based on the first invariant of the DIEM mass matrix is used to determine the optimal parameters of the CB-DIEM scheme, namely the proportionality ratio and number of layers in the DIEM partitioning process and the number of retained frequency modes in the CB reduction method. Furthermore, in implementing the CB method, the inversion of the global stiffness matrix is calculated using only the stiffness matrix of the first element layer. Having reduced the DIEM model, a coupled DIE-FE algorithm is employed to model the dynamic problems of the full structure, which removes the respective methods disadvantages while keeping their advantages. The validity and performance of the proposed CB-DIEM method are investigated by means of three illustrative problems.