Global Stiffness Matrix Research Articles

Locally assembled stiffness matrix: a novel method to obtain global stiffness matrix

Locally assembled stiffness matrix: a novel method to obtain global stiffness matrix

2D Minimum Compliance Topology Optimization Based on a Region Partitioning Strategy

This paper presents an extended sequential element rejection and admission (SERA) topology optimization method with a region partitioning strategy. Based on the partitioning of a design domain into solid regions and weak regions, the proposed optimization method sequentially implements finite element analysis (FEA) in these regions. After standard FEA in the solid regions, the boundary displacement of the weak regions is constrained using the numerical solution of the solid regions as Dirichlet boundary conditions. This treatment can alleviate the negative effect of the material interpolation model of the topology optimization method in the weak regions, such as the condition number of the structural global stiffness matrix. For optimization, in which the forward problem requires nonlinear structural analysis, a linear solver can be applied in weak regions to avoid numerical singularities caused by the over-deformed mesh. To enhance the robustness of the proposed method, the nonmanifold point and island are identified and handled separately. The performance of the proposed method is verified by three 2D minimum compliance examples.

Analysis of Interaction of Multiple Cracks Based on Tip Stress Field Using Extended Finite Element Method

A new method is presented to study the interaction of multiple cracks, especially for the areas near crack tips by using the extended finite element method. In order to track the cracks, a new geometric tracking technique is proposed to track enriched elements and nodes along the crack instead of using the narrow band level set method. This allows to accurately determine enriched elements and nodes and calculate enrichment values. A method is proposed for constructing a multicrack matrix, which involves numbering enriched nodes of multiple cracks and solving the global stiffness matrix. In this approach, the stress fields around multiple cracks can be studied. The interaction integral method is employed to study the crack propagation and its direction by calculating the stress intensify factor. The developed model has been coded in MATLAB environment and validated against analytical solutions. The application of the model in the crack interaction study is demonstrated through a number of examples. The results illustrate the influence of the interaction of multiple cracks as they approach each other.

An Improved Dynamic Model and Matrix Displacement Model for Distributed-Compliance Bridge-Type Amplification Mechanism

This paper establishes a matrix displacement model and an improved dynamic model for the static and dynamic performances analysis for a kind of bridge-type displacement amplification mechanism with distributed-compliance, which has better performances than traditional lumped-compliance bridge-type mechanisms. In the matrix displacement model, the stiffness matrix for two rigid bodies connected by flexures is first obtained by regarding the displacements and the forces on two mass centers of the rigid bodies as the node displacements and node forces. By extending and superimposing each elemental stiffness matrix, the global stiffness matrix for the flexure mechanism can be obtained to calculate the displacement amplification ratio and input stiffness of the bridge-type mechanism. In the improved dynamic model, in order to establish the Lagrangian dynamic model more accurately, the deflectional, axial, and rotational velocities of any point on the beam flexure are calculated by solving the derivatives of the deformation curves of beam flexures versus time to obtain the expression of the kinetic energy in the vibrating beams. On this basis, the three-degree-of-freedom vibration differential equation for the bridge-type mechanism is established by using the Lagrange method, and the natural frequency in the working direction is obtained accurately. The presented models are compared with the finite element analysis, and experiments for two case studies of the bridge-type distributed-compliance mechanism are presented. The comparisons results demonstrate the high prediction accuracy of the improved dynamic model.

Multi-mode propagation and diffusion analysis using the three-dimensional second strain gradient elasticity

The multi-mode propagation and diffusion properties are crucial informations when studying complex waveguides. In this paper, firstly, the three-dimensional modeling of micro-sized structures is introduced by using the second strain gradient theory. The constitutive relation is deduced while the six quintic Hermite polynomial shape functions are employed for the displacement field. The weak formulations including element stiffness and mass matrices and the force vector are calculated through the Hamilton’s principle and the global dynamic stiffness matrix of a unit cell is assembled. Then, free wave propagation characteristics are analyzed by solving eigenvalue problems within the direct wave finite element method framework. The dispersion relations of positive going waves considering the size effects are illustrated. Furthermore, the effects of higher order parameters on the dispersion curves are discussed and the forced responses with two boundary conditions are expounded. Eventually, the wave diffusion including reflection and transmission coefficients are illustrated through simple and complex coupling conditions, respectively. The dynamic analysis of coupled waveguides through the wave finite element method equipped with the second strain gradient is a novel work. The results show that the proposed approach is of significant potential for investigating the wave propagation and diffusion characteristics of micro-sized structures.

A unified approach for relationships among Green's function, normal modes and dispersion spectrum in layered elastic half-space, with corrected misconceptions on surface wave dispersion and testing

SUMMARY In this paper, the global stiffness matrix [K] and the Fourier–Bessel series methods are proposed to derive the accurate Green's function and dynamic response in a form that is directly related to the dispersion curve and experimental dispersion spectrum. Detailed analyses are carried out for the two-layered half-space with different velocity profiles, including the homogeneous half-space as a special case. Our studies indicate that, in Rayleigh wave analysis, the original Rayleigh equation, instead of the rationalized Rayleigh equation as previously derived and used, should be used since the latter would contain extra non-physical roots. We further reveal and characterize three distinct types of leaky waves: the intrinsic surface leaky wave, the apparent Rayleigh mode with a frequency gap associated with a low-velocity half-space and the fast-guided P–SV wave in the layered medium with a high VS contrast between the upper layer and the lower half-space. All leaky modes can be captured by local minima of |det[K]| instead of tracing complex roots in other existing approaches. In the experimental estimation of dispersion curves for practical applications, we have observed that the truncation effect is the major source of uncertainty regardless of the wavefield transformation method utilized. Furthermore, the truncation effect is both location- and model-dependent, without a unique optimal near offset. As such, in order to reduce the uncertainty from the truncation effect, the receiver layout should be considered in the inversion of dynamic response, instead of relying on ensuring a minimum near offset. This becomes possible with the present fast and accurate complete dynamic Green's function by which all wave phenomena (including different types of leaky waves) and receiver locations can be considered in the wavefield transformation.

An improved efficient implicit solution strategy for elastic cracking simulation based on ordinary state-based peridynamics

Peridynamics is a non-local theory based on integral equations to simulate cracks or fracture behaviors. So far, most solution techniques use explicit algorithms with a small timestep, however, leading to low efficiency and accuracy. Conventional implicit strategies, such as Newton-Raphson, can alleviate those disadvantages because the timestep is no longer restricted to a very small value and cumulative error can be eliminated. Although conventional implicit strategies (e.g., Newton-Raphson) can get rid of computation limits with a somehow larger timestep, additional computational efforts are consumed for the formation and decomposition of a high-dimensional global stiffness matrix in the case of a large-scale model. For this reason, a reduced-order Newton-Raphson strategy was proposed in which the global non-linear iteration analysis was replaced with a reduced-order one. In this case, only the small capacitance matrix related to broken bonds needs to be formed and decomposed rather than the global stiffness matrix. Enlightened by this, this study proposed a modified solution strategy for the cracking simulation of elastic bodies based on ordinary state-based peridynamics. In specific, the inversion of the secant stiffness matrix is updated in each nonlinear iteration, during which the capacitance matrix related to new broken bonds is solved. Once the maximum number of the new broken bonds in the iteration is limited, the capacitance can therefore be constrained in a low-dimensional space. Three practical examples using the proposed implicit solution strategy shown in the study demonstrated that the size of the matrix operation can be significantly reduced using the proposed method and accordingly, the computational efficiency is improved.

The V-MVLE model for cyclic failure behavior simulation of planar RC members

As a widely used planar structural component, the reinforced concrete (RC) shear wall in service is usually subjected to axial, bending and shear loading, possessing high dissymmetry damage mechanics of concrete and failure mode of steel bars. To obtain a satisfactory level of simulation accuracy for the cyclic failure behavior of RC shear walls, progresses were made both in microscopic and macroscopic modeling method. In this study, a hybrid planar model based on vector form intrinsic finite element (VFIFE) and macroscopic multi-vertical line element (MVLE) was proposed, which combined the merits of detailed response of the microscopic model and high calculation efficiency of the macroscopic model. Owing to particle-based formulation along the path element, no assembling of a global stiffness matrix was required by the VFIFE method, which attributed the advantages in large deformation and reinforcement fracture behavior analysis for the proposed model during cyclic failure phase. Via the proposed hybrid model of MVLE, the RC shear walls with various parameters, involving axial compression ratios and height–weight​ ratios, were simulated to allow comparison between the experimental and analytical results in reversed cyclic tests, which demonstrated good agreement and satisfied calculation efficiency.

Forward calculation of displacement fields with multilayered unsaturated highway system induced by falling weight deflectometer using dynamic response method

A new dynamic forward model of a multilayered highway structure, which considers the unsaturated characteristics of subgrade soil, is established in this paper to assess the dynamic response induced by the falling weight deflectometer (FWD) load. The Laplace-Hankel transform and dynamic stiffness matrix method are employed to solve the governing equations of the unsaturated soil layer. Combined with the global stiffness matrix and boundary conditions, analytical solutions of the entire system can be derived in the transform domain. A numerical integration method is utilized to perform the Laplace-Hankel inverse transform to obtain time-domain solutions. An analysis of the effects of saturation on the stress, displacement, and pore water pressure is presented. It is revealed that the effect of saturation of unsaturated subgrade soil on vertical stress is negligible, whereas vertical displacement and pore water pressure increase significantly with increasing saturation. Additionally, the vertical displacement response on the pavement surface is sensitive to subgrade deterioration and variations in material properties. Moreover, subgrade deterioration can be reflected in the variation in vertical displacement at approximately 1.2 m from the load center for the studied road structure. The proposed model, which is more realistic and reasonable, can provide a theoretical basis for the back-calculation of the elastic modulus of multilayered highway systems.

Parallelized Element-By-Element Solver for Structural Analysis of Flexible Pipes Using Finite Macroelements

Abstract Due to the number of layers and their respective geometrical complexities, finite element analyses of flexible pipes usually require large-scale schemes with a high number of elements and degrees-of-freedom. If proper precautions are not taken, such as suitable algorithms and numerical methods, the computational costs of these analyses may become unfeasible. Finite macroelements are finite elements formulated for the solution of a specific problem considering and taking advantage of its particularities, such as geometry patterns, to obtain computational advantages, as reduced number of degrees-of-freedom and ease of problem description. The element-by-element (EBE) method also fits very well in this context, since it is characterized by the elimination of the global stiffness matrix and its memory consumption grows linearly with the number of elements, besides being highly parallelizable. Over the last decades, several works regarding the EBE method were published in the literature, but none of them directly applied to flexible pipes. Due to the contact elements between the layers, problems with flexible pipes are usually characterized by very large matrix bandwidth, making the implementation of the EBE method more challenging, so that its efficiency and scalability are not compromised. Therefore, this work presents a parallelized implementation of an element-by-element architecture for structural analysis of flexible pipes using finite macroelements. Four synchronization algorithms were developed and analyzed in detail, including their scalability assessment, and comparisons were made with a well-established finite element method (FEM) software with significant gains in simulation time and memory consumption.

Influence of scour on the vertical response of rock‐socketed piles in layered saturated rock–soil mass

AbstractThe influence of scour on the vertical response of rock‐socketed piles in a layered saturated rock–soil mass is studied through the finite element‐boundary element coupling method. By utilizing the theoretical solution of layered saturated media as the kernel function, the boundary element method (BEM) is used to deduce the boundary integral equations of rock–soil mass. Meanwhile, the pile is modelled with the three‐node bar element and the global stiffness matrix of the pile is assembled by the finite element method (FEM). Then according to the scour type, the flexibility matrix of the foundation is recombined and extended, which is consistent with the stiffness matrix dimension of the pile. In consideration of the force balance and continuity conditions at the pile‐soil‐rock interface, the FEM–BEM coupling equation is established to solve the interaction between layered rock–soil mass and rock‐socketed pile that considers different scour types. A corresponding MATLAB program is compiled to verify the correctness of the present theory by comparisons with the existing literature solutions, field tests and the finite element solutions. The influences of the scour type, scour depth, socketed length as well as pile‐rock mass modulus ratio and rock mass stratification on the vertical response of a rock‐socketed single pile are further discussed through several numerical examples.

Shape optimization with immersed interface finite element method

AbstractShape Optimization using conventional interface‐fitted finite element method requires either mesh morphing or remeshing, which is computationally expensive and prone to errors due to suboptimal mesh or solution interpolation. In this article, we present a shape optimization scheme which uses a four‐noded rectangular immersed‐interface element circumventing the need for interface‐fitted mesh for finite element analysis, and mesh morphing or re‐meshing for shape optimization. The analysis problem is solved using a non‐conformal, Petrov–Galerkin (ncPG) formulation. We use non‐conformal trial functions with conformal test functions. A fixed structured grid with a linear approximation for the interface within each element is used. We perform a bi‐material patch test to confirm the consistency and convergence of the immersed‐interface finite element method (IIFEM). The convergence of the displacement and stress error norms are of the same or slightly better order compared to the interface‐fitted FEM. Thus, IIFEM reduces the cost of meshing without compromising the accuracy of the solutions obtained. We then perform analytical design sensitivity analysis to obtain the gradient of the global stiffness matrix with respect to shape design variables. The sensitivities are used in the gradient‐based optimization formulation of the shape design problem. We present several ways to parametrize the design space. Finally, verification and case studies are presented to demonstrate the accuracy and potential of the approach.

Exact Matrix Stiffness Method for Out-of-Plane Buckling Analysis of Funicular Arches Considering Warping Deformations

The out-of-plane buckling behavior of arches is closely related to the element torsional behavior. The traditional 12-degree-of-freedom second-order element stiffness matrix which uses a simplified element torsional stiffness GJ/[Formula: see text] (where [Formula: see text] is the shear modulus, [Formula: see text] the St. Venant torsion constant, [Formula: see text] the element length) may significantly underestimate the out-of-plane buckling loads of funicular arches. This paper presents a simple and effective exact matrix stiffness method (MSM) for the out-of-plane buckling analysis of funicular arches. The developed MSM uses a 14-degree-of-freedom second-order element stiffness matrix of three-dimensional beam-columns considering both torsion and warping deformations. The out-of-plane buckling analysis of funicular arches is performed by using the global structural stability stiffness matrix, which combines the transformed second-order element stiffness matrices. The proposed MSM with the exact 14-degree-of-freedom second-order element stiffness matrix for the out-of-plane buckling analysis is verified by comparing with some classical solutions of funicular circular and parabolic arches with box sections and I-sections. Further discussions show that the 14-degree-of-freedom second-order element stiffness matrix may be reduced to a simplified 12-degree-of-freedom form only by deriving the exact element torsional stiffnesses, which could be significantly larger than GJ/[Formula: see text] for members with large cross-sectional torsional stiffness parameters (especially open cross-sections).

Seismic and failure behavior simulation for RC shear walls under cyclic loading based on vector form intrinsic finite element

Seismic performance and failure mode simulations for reinforced concrete (RC) structures are impeded by large deformations and material nonlinearity particularly when subjected to the combined action of compression, bending and shear. Structural deformation and failure analysis based on traditional finite element (FE) often fails for the small–deformation assumption and global equilibrium. Moreover, it remains difficult to obtain convergence results for the existing FE methods by applying general concrete constitutive models for the complicity of anisotropic materials. To simulate the hysteretic performance and the damage development process of the RC shear wall under cyclic loading, the vector form intrinsic finite element (VFIFE) method with independent particles was adopted and combined with the two-directional concrete constitutive model considering the loading and unloading rules to perform large deformation and damage estimations. Concrete planar triangular element and fiber element for steel bar equipped with the hysteretic constitutive models were used to model six RC shear walls with different design parameters. Owing to the merits of avoiding repeated accumulation of the global stiffness matrix for vector mechanics, the planar structural behaviors involving biaxial stress–strain and loading–unloading behavior were numerically demonstrated. The simulated results were in close agreement with the experimental tests, including the hysteretic loop, skeleton curve, bending–shear deformation, and damage mode. This research also provides successive evidence for damage process simulation by VFIFE for RC structures under seismic loading considering a comprehensive constitutive relation.

Static, free vibrational and buckling analysis of laminated composite beams using isogeometric collocation method

Isogeometric collocation (IGA-C) method is a computational approach to solve boundary value problems. In this method (IGA-C), the differential equations are solved in strong form instead of the weak form approach adopted by Galerkin based formulations. IGA-C method is computationally efficient in comparison to conventional finite element method and Galerkin-Isogeometric approaches. IGA-C method does not involve the process of assembling global stiffness matrix from element stiffness matrix. Another advantage of IGA-C is that it requires a single integration point per element irrespective of the order of Non-Uniform Rational B-Spline functions (NURBS) adopted. Isogeometric collocation has also been demonstrated as a stable, efficient and accurate higher order computational method for explicit problems. For a wider adoption of isogeometric collocation method, beam/plate/shell finite elements within the framework of IGA-C method need to be formulated. Owing to the extensive adoption of laminated composites in structural components, development of beam finite elements for laminated composite beams based on isogeometric collocation method will prove useful during analysis of composite structures. IGA-C method is proposed in this study for the static bending, free vibration and buckling analysis of laminated composite beams. Classical laminated plate theory (CLPT), first order shear deformation theory (FSDT) and higher order shear deformation theory (HSDT) are considered for all the three analyses. The computational approach proposed for laminated beam based on HSDT contains two Degrees of Freedom (DOF) per node. Computational approach for analysing laminated composite beams based on each of these kinematic theories and using IGA-C method is presented. Accuracy of the proposed computational approaches is checked by solving different numerical examples. Values of normalized transverse displacement, normalized stresses, normalized natural frequencies and normalized critical buckling loads are compared with results from the literature and are found to be accurate.

Influence of the Order Exchange of the Node Connection in the Force Analysis of Steel Structures

In the mechanical analysis of steel structures, whether it is static analysis or dynamic analysis, it is necessary to establish the structural stiffness matrix first. In the process of building the structural stiffness matrix, the same element usually has different node code connection orders, and it has never been argued whether the different connection orders of the same element will have an effect on the building of the stiffness matrix. In this study, the influence of the difference in the node connection order on the construction of the element stiffness matrix is studied. First, the structural element stiffness matrix in the global coordinate system is established when the node connection order is different. It is found that the element stiffness matrix in the global coordinate system is indeed inconsistent for the same element with different connection orders. In this study, the elements of the established element stiffness matrix are extracted into the global stiffness matrix of the structural system based on the law of energy conservation; it is found that the global stiffness matrix finally established by using two different connection relationships is the same. The research results of the example show that in the stress analysis of steel structures, selecting different node connection sequences to establish the structural stiffness matrix will obtain the element stiffness matrix under different global coordinate systems. However, through the aggregation process of the global stiffness matrix of the structural system, the global stiffness matrix obtained is consistent, so the different connection sequences of nodes will not affect the stress analysis of steel structures. The example further analyzes the static stress and dynamic responses of the steel structure. The conclusions of this study provide a reliable theoretical basis for the situation that the order of node connections need not be consistent in the finite element modeling of steel structures and are of reference value for the finite element modeling of steel structures.

A Structural Damage Identification Method Based on Arrangement of the Static Force Residual Vector

In order to effectively and conveniently identify the damage location and damage degree of structural members under static response, a structural damage identification method based on the force residual vector is proposed. The force residual vector is defined by using the static displacement data and the stiffness matrix of the finite element model structure. The structural element corresponding to the non-zero element of the permutation force residual vector is intelligently determined as the damage element. The damage degree of the damaged unit is calculated from the equilibrium equation, which is established by the global stiffness matrix with only the damaged unit. For example, the identification analysis of damage units of numerical models is carried out for a simply supported beam as a simple structure and a truss as a complex structure based on the proposed method. In El Centro seismic wave, the dynamic responses of the original model and truss damage model are simulated and compared by using state space theory to verify the necessity of damage identification.

A physics-guided machine learning for multifunctional wave control in active metabeams

With the growing interest in the field of artificial materials, more advanced and sophisticated wave functionalities are required from phononic crystals and acoustic metamaterials. Due to expensive iterative fitness evaluations, inverse designing acoustic metamaterials with desired physical responses using conventional optimization approaches remains challenging especially in high-dimensional design space. To address this issue, we suggest a physics-guided machine-learning-based inverse design approach for realizing multifunctional wave control in active metabeams connecting with negative capacitances (NCs). The transfer matrix method which relates the wave field and its derivative to carry the fundamental wave propagation information will be embedded in the ML network to construct the complex mapping between the input and output responses of the unit cell. After this network is well trained, global wave propagation behavior in the active metabeam can be accurately described by the concatenation of networks of each unit cells into a global stiffness matrix. After the performance of the network is validated by conducting numerical simulation, we further apply the proposed network as a surrogate model for genetic algorithm on the inverse design of the metabeam for quick realization multifunctional wave control abilities without changing microstructures. Our proposed approach can not only be easily extended to design other types of active/passive metamaterials, but also provides some insights into optimization aided engineering in high-dimensional (2D and 3D) design space of active metamaterials.

A single step optimization method for topology, size and shape of trusses using hybrid differential evolution and symbiotic organisms search

In this article, a single step optimization approach for topology, size and shape of trusses subjected to multiple static and free vibration constraints is developed. For that aim, a topology pseudo-area variable based on a penalty parameter is discretely assigned by either 10−3 or 1 which represents the absence or attendance of a truss member. This helps to not only dodge the singularity of global stiffness matrix as resolving the equilibrium equation system, but also preserve the intact finite element model structure without unnecessarily and repeatedly done time-consuming performances in finite element analyses. The members’ cross-sectional area serves as discrete/continuous size variables, whilst nodes’ spatial coordinates are considered as continuous shape ones. The structural weight is minimized with multiple restrictions on kinematic stability, stress, displacement, natural frequency and Euler buckling loading. A hybrid differential evolution and symbiotic organisms search is employed as an optimizer and refined to tackle both continuous and discrete variables. Eight well-known examples for simultaneous topology, size and shape optimization of 2D and 3D trusses imposed by multiple static and free vibration constraints are tested to verify the reliability and the robustness of the suggested paradigm. The current method can result in competitive high-quality solutions against other state-of-the-art algorithms in most of the examined examples. In addition, the current approach can be also extended to apply for simultaneous topology, size and shape optimization of large-scale trusses in practice.

Vibration Characteristic Analysis of Tunnel Structure Buried in Elastic Foundation Based on Dynamic Stiffness Matrix

In this study, the free vibration characteristics of a non-circular tunnel buried in a Winkler foundation are first investigated based on a dynamic stiffness approach and the Wittrick–Williams algorithm. The non-circular tunnel was modeled as a shell composed of multipartial circular cylinders based on the first shear deformation Flügge shell theory, by which the effects of shear deformation and moment of inertia were considered. The transfer function of the state variable was derived, and the elementary dynamic stiffness matrices were formulated based on this transfer function. Similar to the finite element method, the global dynamic stiffness matrix was established by assembling each elementary dynamic stiffness matrix. By solving the global dynamic stiffness matrix using the Wittrick–Williams algorithm, the natural frequencies and the corresponding mode shapes were determined. Subsequently, the calculated results were compared with literature and finite element analysis results, thus validating the accuracy and reliability of the proposed method. A parametric analysis was conducted to investigate the effects of the cross-section, stiffness coefficient, thickness, and length of the tunnel composed of multipartial circular cylinders. It was concluded that among the four types of cross-sections, the free vibration characteristics of the shells with close isoperimetric radii are similar. The natural frequencies of the shells buried in a moderately stiff Winkler foundation increase rapidly with its stiffness coefficient, whereas those of the shells buried in softer and stiffer foundations remain practically constant. The increase in shell thickness progressively increases the natural frequencies. In contrast, the natural frequencies decrease sharply with the increase in shell length; they first decrease to the lowest limits and subsequently remain unaltered. Thus, the free vibration characteristics of an actual tunnel structure resting on a Winkler foundation can be obtained by conducting the free vibration analysis on a sufficiently long shell segment, which is of major significance in the dynamic design of tunnel structures.