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Showing papers on "Tangent stiffness matrix published in 2020"


Journal ArticleDOI
TL;DR: In this article, a nonlocal operator method is proposed which is generally applicable for solving partial differential equations (PDEs) of mechanical problems, which can be regarded as the integral form "equivalent" to the differential form in the sense of nonlocal interaction model for solving the unknown field.

130 citations


Journal ArticleDOI
TL;DR: In this article, the rate-problem of continuing equilibrium for a general class of rate-independent elastoplastic solids, without assuming the normality flow rule or symmetry of the tangent stiffness matrix, is examined.
Abstract: The rate-problem of continuing equilibrium is examined for a general class of rate-independent elastoplastic solids, without assuming the normality flow rule or symmetry of the tangent stiffness matrix. Accordingly, the problem addressed is of non-potential type, for which the usual stationarity or minimum principles for a governing potential do not apply. It is shown that the rate-problem can nevertheless be formulated as a quasi-extremal energy principle. It is characterized by explicit dependence of the minimized energy function or functional not only on variables undergoing variations but also, although only in a particular way, on an unknown solution as a parameter. To enable transparent and mathematically simple presentation of the main concept, the energy function is defined in a finite-dimensional setting for a spatially discretized material body with generalized velocities and a number of plastic multipliers as unknowns. If a solution is non-unique then incrementally stable solutions can be selected using the quasi-extremal principle in which the minimized energy function includes the second-order terms. Examples and extensions concern an elastic-plastic continuum obeying a non-associative plastic flow rule, without or with a higher-order gradient term in the loading function. The issue of selection of active slip-systems in a single crystal of a non-symmetric slip-system interaction matrix is also addressed.

14 citations


Journal ArticleDOI
Gang Li1, Jia-Long Li1, Long Yu1, Ding-Hao Yu1, Zhi-Qian Dong1 
TL;DR: In this paper, an inelasticity-separated solid element model and an efficient numerical solution procedure are proposed for the material nonlinear analysis of three-dimensional (3D) entity structures.

8 citations


Journal ArticleDOI
TL;DR: In this article, the authors present a non-linear finite element method with nonlinear material models, where stress and stiffness fields, as well as displacements at each degree of freedom, are modeled as random processes and expanded along the Polynomial Chaos (PC).

8 citations


Journal ArticleDOI
TL;DR: This paper uses automatic differentiation for stating and solving solid mechanics problems, and makes use of AD for directly obtaining the residual force vector and the tangent stiffness matrix of the problem, as the gradient and the Hessian of the free energy respectively.
Abstract: Automatic differentiation (AD) is an ensemble of techniques that allow to evaluate accurate numerical derivatives of a mathematical function expressed in a computer programming language. In this paper we use AD for stating and solving solid mechanics problems. Given a finite element discretization of the domain, we evaluate the free energy of the solid as the integral of its strain energy density, and we make use of AD for directly obtaining the residual force vector and the tangent stiffness matrix of the problem, as the gradient and the Hessian of the free energy respectively. The result is a remarkable simplification in the statement and the solution of complex problems involving non trivial constraints systems and both geometrical and material non linearities. Together with the continuum mechanics theoretical basis, and with a description of the specific AD technique adopted, the paper illustrates the solution of a number of solid mechanics problems, with the aim of presenting a convenient numerical implementation approach, made easily available by recent programming languages, to the solid mechanics community.

8 citations


Journal ArticleDOI
TL;DR: In this paper, the relationship between members' stiffness and structural stiffness is established, and the structural demand stiffness is extracted from the overall stiffness of the structure by finding the members that contribute the most to resist an external load.

8 citations


Journal ArticleDOI
TL;DR: In this paper, a numerical algorithm combining the meshless collocation technique based on the globally supported Multiquadric Radial Basis Function (MQRBF) approximation method and the Asymptotic Numerical Method (ANM) is developed to investigate the nonlinear and linear static behaviors of Single-Walled Carbone Nanotubes (SWCNTs) based on a Nonlocal Continuum Beam Model (NCBM).
Abstract: In this paper, a numerical algorithm combining the meshless collocation technique based on the globally supported Multiquadric Radial Basis Function (MQRBF) approximation method and the Asymptotic Numerical Method (ANM) is developed to investigate the nonlinear and linear static behaviors of Single-Walled Carbone Nanotubes (SWCNTs) based on a Nonlocal Continuum Beam Model (NCBM). The NCBM which accounts for the effects of small scale, shear deformation and geometric nonlinearity of von-Karman type is constructed by implementing the Nonlocal Elasticity Theory of Eringen (NET) into the First-Order Shear Deformation Elastic Beam Theory (FOSDEBT) and used to model the SWCNTs as a Continuous Elastic Nanobeam (CEN). Based on the Total Lagrangian Formulation (TLF) the nonlocal nonlinear governing equations of SWCNT are elaborated in a quadratic matrix strong form. These equations are then solved numerically by applying the proposed algorithm. This algorithm permits to transform the quadratic nonlocal nonlinear governing equations into a sequence of linear ones shared the same tangent stiffness matrix which are then solved numerically by the MQRBF approximation method, to get the whole branch of solution a continuation procedure is required. In order to highlight the reliability and efficiency of the present numerical algorithm, comparative studies are conducted.

8 citations


Journal ArticleDOI
TL;DR: The co-rotational method is used to solve four benchmark problems from the literature, including optimizing for stiffness, compliant mechanism design, and a plate problem, demonstrating the potential of the co- rotational method as an alternative approach for geometrically nonlinear topology optimization.
Abstract: This paper investigates the application of the co-rotational method to solve geometrically nonlinear topology optimization problems. The main benefit of this approach is that the tangent stiffness matrix is naturally positive definite, which avoids some numerical issues encountered when using other approaches. Three different methods for constructing the tangent stiffness matrix are investigated: a simplified method, where the linear elastic stiffness matrix is simply rotated; the consistent method, where the tangent stiffness is derived by differentiating residual forces by displacements; and a symmetrized method, where the consistent tangent stiffness is approximated by a symmetric matrix. The co-rotational method is implemented for 2D plane quadrilateral elements and 3-node shell elements. Matlab code is given in the appendix to modify an existing, freely available, density-based topology optimization code so it can solve 2D problems with geometric nonlinear analysis using the co-rotational method. The approach is used to solve four benchmark problems from the literature, including optimizing for stiffness, compliant mechanism design, and a plate problem. The solutions are comparable with those obtained with other methods, demonstrating the potential of the co-rotational method as an alternative approach for geometrically nonlinear topology optimization. However, there are differences between the methods in terms of implementation effort, computational cost, final design, and objective value. In summary, schemes involving the symmetrized tangent stiffness did not outperform the other schemes. For problems where the optimal design has relatively small displacements, then the simplified method is suitable. Otherwise, it is recommended to use the consistent method, as it is the most accurate.

7 citations


Journal ArticleDOI
TL;DR: In this paper, the proper orthogonal decomposition (POD) is incorporated into the spectral stochastic FEM with local basis functions in the Stochastic domain (SL-FEM), thus performing a drastic reduction of the stiffness matrix.

5 citations


Journal ArticleDOI
TL;DR: In this paper, a nonlinear form-finding method for symmetric cable-strut structures with complex geometry or many nodes is proposed, where the first block matrix of the symmetry-adapted tangent stiffness matrix is extracted using the full symmetry subspace, which is associated with the first irreducible representation of a symmetry group.
Abstract: An efficient and nonlinear form-finding method is proposed for symmetric cable–strut structures with complex geometry or many nodes. Expressed in the symmetry-adapted coordinate system, the first block matrix of the symmetry-adapted tangent stiffness matrix is extracted using the full symmetry subspace, which is much smaller-sized and associated with the first irreducible representation of a symmetry group. Then, this stiffness submatrix and the principle of minimum potential energy are adopted for the fast but stable convergence of the initial configuration to the stable configuration. During the form-finding process, the generalized inverse of a matrix and modification for the minimum eigenvalues are employed, to guarantee the positive definiteness of the stiffness submatrix. The form-finding process can start from an arbitrary initial configuration, whereas only certain symmetry group, and the connectivity pattern and the initial lengths of the members should be given in advance. A few numerical examples are illustrated to show the efficiency and accuracy of the form-finding method for cable–strut structures with complex geometry and different symmetry.

3 citations


Journal ArticleDOI
Linzi Fan1, Yue Sun2, Weiyin Fan2, Yao Chen2, Jian Feng2 
TL;DR: In this paper, a computational method is proposed for seeking zero-stress states of symmetric cable-strut structures by evaluating distributed static indeterminacy and symmetry representations using group theory, the active member with proper importance index and high-order symmetry is chosen from different types of members.
Abstract: A prestressed cable–strut structure is generally flexible and exhibits strong coupling between its stress state and configuration. The zero-stress state offers the basis for design and analysis of cable–strut structures and has significant influence on the prestress state and the load state. Here, a computational method is proposed for seeking zero-stress states of symmetric cable–strut structures. By evaluating distributed static indeterminacy and symmetry representations using group theory, the active member with proper importance index and high-order symmetry is chosen from different types of members. Moreover, natural lengths and the involved elongations of the members are established from the initial prestresses and geometric properties. Then, based on the Newton method and the Moore–Penrose inverse theory, internal forces of the members are actively reduced. The structural configuration and tangent stiffness matrix are iteratively updated during the whole process from the prestress state to the zero-stress state. The feasibility and accuracy of the proposed approach are verified by some numerical examples, whereas the results are compared with analytical solutions and FEM simulation. The results show that one zero-stress configuration is associated with a specific prestress state, and the process between zero-stress state and prestress state is reversible. This work has theoretical significance for the design of novel cable–strut structures and provides a reference for the construction process of prestressed cable–strut structures in practical applications.

Journal ArticleDOI
TL;DR: The stability of borehole is a major concern in petroleum and geotechnical engineering as mentioned in this paper, and the stability of the borehole has been studied extensively in the past few decades.
Abstract: The stability of borehole is a major concern in petroleum and geotechnical engineering. Subsurface fossil fuel and thermal energy extraction, deep geologic carbon/energy storage, and waste ...

Journal ArticleDOI
TL;DR: The results demonstrated that the developed nonlinear multiscale formulation for the 3D problems could provide high precision solutions as well as acceptable numerical efficiencies.
Abstract: An efficient three-dimensional (3D) multiscale method has been introduced to simulate the geometrically nonlinear behaviors of the plant inspired smart cellular structures. In this method, the scale gap between the geometrical information of motor cells in the small-scale and mechanical behaviors of the cellular structures at the macroscale is bridged through a multiscale framework named multiscale finite element method. The heterogeneous information of the microstructure is then equivalent to the macroscopic coarse elements through the multiscale base functions about the displacements for the solid matrix as well as the fluid pressure. Combined with the “element-independent” corotational algorithm, both the tangent stiffness matrix of the coarse grid elements and their nodal forces can be directly deduced, which will be utilized to decompose the geometrically nonlinear motions of equivalent coarse grid elements at the macroscale level. Consequently, the initial geometrically nonlinear behaviors of the 3D fluidic cellular structures could be simulated by the iteration procedures on the coarse-grid meshes, which will greatly reduce the computation time and memory cost. At the same time, the mechanical responses of the motor cells in the microscale could be easily computed from the obtained macroscopic solutions by the downscaling technique of the multiscale method. To verify the proposed nonlinear multiscale method, some numerical examples are presented. The results demonstrated that the developed nonlinear multiscale formulation for the 3D problems could provide high precision solutions as well as acceptable numerical efficiencies.

Proceedings ArticleDOI
25 Nov 2020
TL;DR: An effective and flexible way to assemble tangent stiffness matrices in MATLAB is proposed that is also efficient for purely elastic problems.
Abstract: We propose an effective and flexible way to assemble tangent stiffness matrices in MATLAB. Our technique is applied to elastoplastic problems formulated in terms of displacements and discretized by the finite element method. The tangent stiffness matrix is repeatedly assembled in each time step and in each iteration of the semismooth Newton method. We consider von Mises and Drucker-Prager yield criteria, linear and quadratic finite elements in two and three space dimensions. Our codes are vectorized and available for download. Comparisons with other available MATLAB codes show, that our technique is also efficient for purely elastic problems. In elastoplasticity, the assembly times are linearly proportional to the number of integration points.

Journal ArticleDOI
TL;DR: In this article, a fully coupled poroelastoplasticity reservoir model based on Drucker-Prager yield criterion is implemented the tangent stiffness method, and the computational efficiency is compared with the constant stiffness method.

Posted Content
TL;DR: In this article, a three-dimensional constitutive model for shape memory alloys considering the TRsformation-Induced Plasticity (TRIP) as well as the Two-Way Shape Memory Effect (TWSME) through a large deformation framework is presented.
Abstract: This work presents a three-dimensional constitutive model for shape memory alloys considering the TRansformation-Induced Plasticity (TRIP) as well as the Two-Way Shape Memory Effect (TWSME) through a large deformation framework. The presented logarithmic strain based model is able to capture the large strains and rotations exhibited by SMAs under general thermomechanical cycling. By using the martensitic volume fraction, transformation strain, internal stress, and TRIP strain tensors as internal state variables, the model is capable to capture the stress-dependent TRIP generation when SMAs are subjected to a multiaxial stress state, as well as the TWSME for thermomechanically trained SMAs under load-free conditions. A detailed implementation procedure of the proposed model is presented through a user-defined material subroutine within a finite element framework allowing for solving different Boundary Value Problems (BVPs). Comprehensive instruction on calibrating the model parameters as well as the derivation of continuum tangent stiffness matrix are also provided. In the end, the simulated cyclic pseudoelastic and actuation responses by the presented model for a wide range of SMA material systems under both uniaxial and multiaxial stress states are compared against experimental results to validate the proposed modeling capabilities.