Tangent Matrix Research Articles

Determination of the plastic J integral of ductile material using the XFEM with only Heaviside function and variable-node elements

In this paper, the extended finite element method (XFEM) with only Heaviside function is proposed for the elastic–plastic fracture mechanics (EPFM) modeling. The proposed method removes tip enrichment functions depending on the polar coordinates at the crack front, and a step function only determined by the level set function is utilized to track the crack front. To alleviate the volumetric locking phenomena caused by the plastic incompressibility, the B-bar method is incorporated into the three dimensional (3D) XFEM program. Therefore, the fully integration scheme can be chosen to ensure the accuracy when addressing large plastic deformation in EPFM analysis. Additionally, the material tangent stiffness matrix of Ramberg-Osgood constitutive is given, and the local refinement technique using variable-node elements is adopted to reduce the number of elements and nodes for efficient analysis. A Newton-Raphson iterative algorithm is developed to solve the nonlinear algebraic equations caused by material nonlinearity. Several numerical examples including the determination of crack opening displacement, and the fully plastic J integral in the ductile materials are presented to test the performance of the proposed method. Comparisons with the results from the existing methodologies show that the new enrichment scheme can save computational cost and obtain sufficient accuracy even in the case of 3D curved crack.

Fast implicit update schemes for Cahn–Hilliard-type gradient flow in the context of Fourier-spectral methods

This work discusses a way of allowing fast implicit update schemes for the temporal discretization of phase-field models for gradient flow problems that employ Fourier-spectral methods for their spatial discretization. Through the repeated application of the Sherman–Morrison formula we provide a rule for approximations of the inverted tangent matrix of the corresponding Newton–Raphson method with a selectable order. Since the representation of this inversion is exact for a sufficiently high approximation order, the proposed scheme is shown to provide a fixed-point-type iterative solver for gradient flow problems that require the solution of linear systems in the context of an implicit time-integration. While the proposed scheme is applicable to general gradient flow phase-field models, we discuss the scheme in the context of the Cahn–Hilliard equation, the Swift–Hohenberg equation, and the phase-field crystal equation for which we demonstrate the performance of the proposed method in comparison with classical solvers.

Computationally-efficient locking-free isogeometric discretizations of geometrically nonlinear Kirchhoff–Love shells

Discretizations based on the Bubnov-Galerkin method and the isoparametric concept suffer from membrane locking when applied to Kirchhoff–Love shell formulations. Membrane locking causes not only smaller displacements than expected, but also large-amplitude spurious oscillations of the membrane forces. Continuous-assumed-strain (CAS) elements were originally introduced to remove membrane locking in quadratic NURBS-based discretizations of linear plane curved Kirchhoff rods (Casquero et al., CMAME, 2022). In this work, we propose CASs and CASns elements to overcome membrane locking in quadratic NURBS-based discretizations of geometrically nonlinear Kirchhoff–Love shells. CASs and CASns elements are interpolation-based assumed-strain locking treatments. The assumed strains have C0 continuity across element boundaries and different components of the membrane strains are interpolated at different interpolation points. CASs elements use the assumed strains to obtain both the physical strains and the virtual strains, which results in a global tangent matrix which is a symmetric matrix. CASns elements use the assumed strains to obtain only the physical strains, which results in a global tangent matrix which is a non-symmetric matrix. To the best of the authors’ knowledge, CASs and CASns elements are the first assumed-strain treatments to effectively overcome membrane locking in quadratic NURBS-based discretizations of geometrically nonlinear Kirchhoff–Love shells while satisfying the following important characteristics for computational efficiency: (1) No additional unknowns are added, (2) No additional systems of algebraic equations need to be solved, (3) The same elements are used to approximate the displacements and the assumed strains, (4) No additional matrix operations such as matrix inversions or matrix multiplications are needed to obtain the stiffness matrix, and (5) The nonzero pattern of the stiffness matrix is preserved. Analogously to the interpolation-based assumed-strain locking treatments for Lagrange polynomials that are widely used in commercial FEA software, the implementation of CASs and CASns elements only requires to modify the subroutine that computes the element residual vector and the element tangent matrix. The benchmark problems show that CASs and CASns elements, using either 2 × 2 or 3 × 3 Gauss–Legendre quadrature points per element, are effective locking treatments since these element types result in more accurate displacements for coarse meshes and excise the spurious oscillations of the membrane forces.

Analytical Linearization of Aerodynamic Loads in Unsteady Vortex-Lattice Method for Nonlinear Aeroelastic Applications

This paper presents the analytical linearization of aerodynamic loads (computed with the unsteady vortex-lattice method), which is formulated as tangent matrices with respect to the kinematic states of the aerodynamic grid. The loads and their linearization are then mapped to a nonlinear structural model by means of radial-basis functions, allowing for a two-way strong interaction scheme. The structural model comprises geometrically exact beams formulated in a director-based total Lagrangian description, circumventing the need for rotational degrees of freedom. The structural model is spatially discretized into finite elements and temporally discretized with the help of an implicit scheme that identically preserves momenta and energy. The resulting nonlinear discrete equations are solved by applying Newton’s method, requiring calculating the Jacobians of the whole aeroelastic system. The correctness of the linearized loads is then shown by direct comparison with their numerical counterparts. In addition, we employ our strongly coupled aeroelastic model to investigate the nonlinear static and dynamic behavior of a suspension bridge. With this approach, we successfully investigate the numerical features of the aeroelastic system under divergence and flutter conditions.

The reduced order model for creep using dynamic mode decomposition

Given the computational challenges associated with finite element methods in analyzing creep behavior in high-temperature components, this work presents an advanced method for modeling thermal creep in complex structures through a reduced order model (ROM) developed via dynamic mode decomposition (DMD). The results indicate that the ROM using DMD can predict long-term structural responses related to creep based on short-term data with high accuracy, enhancing significantly computational efficiency. The application of ROM allows for the rapid estimation of creep effects in complex structures, like monoliths in the MegaPower reactor, thereby improving the design efficiency. The eigenvalues of the creep related ROM all fall in the unit circle on the complex plane, indicating that creep is a stable development process. The DMD with improved time mode coefficient can reconstruct strain-rate related plastic deformation and cyclic visco-plastic deformation process. Additionally, we derive the necessary stress updating algorithm and consistent tangent operator matrix for the 316H unified visco-plastic constitutive equation in 2023 edition ASME code, ensuring accurate modeling of material creep behavior under high temperatures in this paper. This work provides a valuable tool for the safe design and operation of critical components in nuclear engineering and other high-temperature applications.

Multiscale enhanced non‐uniform transformation field analysis

AbstractEnhanced transformation field analysis (E‐TFA), recently proposed for reduced‐order modeling, is here formulated for and applied to multiscale analysis. The approach is able to reproduce a highly complex nonlinear macroscale behavior, resulting from a microstructure with cohesive interfaces embedded in an elasto‐plastic bulk. E‐TFA features a consistent tangent matrix in its solution procedure, which enables a straightforward definition of the upscaled tangent stiffness tensor. Numerical tests show that, compared to FE, the proposed approach yields accurate solutions at a lower computational cost.

Research on Intelligent Monitoring System of Intelligent Transmission Line Based on Computer Internet of Things Technology

The author proposes an intelligent early-warning system for preventing external damage of transmission lines. The system uses TI-TMS320DM6446 as the core, the peripheral circuit configuration and video monitoring; The moving objects captured by PTZ camera are detected and identified by using mixed Gaussian model, background difference and minimum tangent matrix, and alarm signals are provided. According to the particularity of the power supply of the equipment on the transmission line, the basic power management of the system is analyzed in detail. A reliable power management scheme is designed. The practical application proves that this method has good real-time performance and has obvious effect on preventing power cable damage by force.

Enhanced transformation field analysis for reduced-order modeling of problems with cohesive interfaces

We propose a novel transformation field analysis (TFA) technique designed to solve problems with cohesive interfaces. The TFA approach, originally proposed for homogenization purposes, is (i) extended to account for arbitrary boundary conditions (i.e. not only those suitable for homogenization), (ii) endowed with an iterative Newton–Raphson solution scheme involving the consistent tangent matrix and (iii) provided with an arc-length strategy to follow possible snap-back branches. The extension to arbitrary boundary conditions leads to an efficient reduced-order modeling approach for solving complex non-linear problems. Numerical tests demonstrate that the approach performs well in challenging scenarios with several interfaces, and that sharp and multiple snap-back branches can be followed. The comparison with finite element results shows that the proposed technique yields fast yet accurate solutions.

A general DBEM for mixed-mode cohesive crack problems

This work proposes a general dual boundary element method (DBEM) for cohesive mixed-mode crack problems. Local cohesive stiffnesses are defined to yield the boundary element nonlinear equations for any mixed-mode cohesive model in a direct way. The main novelty resides on the fact that the formulation can take into account all possible cohesive surface conditions: (i) contact, (ii) softening, (iii) unloading/reloading and (iv) complete failure, which may occur independently of the chosen cohesive model. Besides, the well-established Park–Paulino–Roesler (PPR) model was incorporated into DBEM methods. The tangent matrix of the discrete model is explicitly derived and the solution is sought by a proper degree-of-freedom (DOF) control method. The proposed formulation is validated reproducing mode I and mode II patch-tests. Some other relevant problems, including a mixed-mode fracture test, are also presented to demonstrate that the above conditions are essential to reproduce complex mixed-mode cohesive fractures. The results also shown that the control of a monotonically increasing DOF yields reliable solutions in problems with snap-back instabilities. In order to facilitate the validation of the method, predefined crack paths were assumed in the examples. Further developments may include incorporating arbitrary crack growth and damage-based cohesive models for fatigue crack growth analysis.

A progressive damage model for 3D woven composites integrating kinking-buckling competition

In this work, a progressive damage model (Kinking-Buckling PDM) considering fiber micro-buckling failure, fiber kinking failure, transverse inter-fiber failure of yarns, matrix failure and interface debonding, is proposed to predict the compressive mechanical behavior of 3D woven composites (3DWC). Fiber kinking and fiber micro-buckling failure are adopted in the form of competition to characterize the longitudinal failure of fiber yarns. A bilinear damage evolution model is established to control the damage accumulation of each constituent. Meanwhile, a material tangent constitutive tensor characterizing the constitutive behavior of 3DWC is derived. The complex failure process of 3DWC under compression is displayed through Kinking-Buckling PDM. As the fiber initial misalignment angle increases, the compressive strength of 3DWC declines significantly, meanwhile, the main failure mode of 3DWC will be transformed from fiber micro-buckling to fiber kinking in warp yarns. The relevant experimental results are used to confirm the accuracy of the PDM.

A numerical framework for simulating progressive failure in composite laminates under high-cycle fatigue loading

In this work, a recently proposed high-cycle fatigue cohesive zone model, which covers crack initiation and propagation with limited input parameters, is embedded in a robust and efficient numerical framework for simulating progressive failure in composite laminates under fatigue loading. The fatigue cohesive zone model is enhanced with an implicit time integration scheme of the fatigue damage variable which allows for larger cycle increments and more efficient analyses. The method is combined with an adaptive strategy for determining the cycle increment based on global convergence rates. Moreover, a consistent material tangent stiffness matrix has been derived by fully linearizing the underlying mixed-mode quasi-static model and the fatigue damage update. The enhanced fatigue cohesive zone model is used to describe matrix cracking and delamination in laminates. In order to allow for matrix cracks to initiate at arbitrary locations and to avoid complex and costly mesh generation, the phantom node version of the eXtended finite element method (XFEM) is employed. For the insertion of new crack segments, an XFEM fatigue crack insertion criterion is presented, which is consistent with the fatigue cohesive zone formulation. It is shown with numerical examples that the improved fatigue damage update enhances the accuracy, efficiency and robustness of the numerical simulations significantly. The numerical framework is applied to the simulation of progressive fatigue failure in an open-hole [±45]-laminate. It is demonstrated that the numerical model is capable of accurately and efficiently simulating the complete failure process from distributed damage to localized failure.

A contact formulation using local frame of SE(3) group for corotational beam against rigid solid via moving signed distance field

In this paper, we present a contact formulation for corotational beam and rigid body based on body-attached signed distance field (SDF). All formulations are based on the unified SE(3) framework, whose prominent advantage lies in its frame invariance property. For solid body, the geometry surface can be implicitly described by a moving SDF. Consequently, the interaction between beams and rigid solids can be viewed as immersion in a field-described force. The contact force is formulated using SDF interpolation, and exact tangent matrices are provided to facilitate implicit iteration. Notably, this approach eliminates the need for contact detection during simulation, because the SDF database is constructed before the simulation. To enhance the spatial adaptivity of the SDF algorithm, we introduce an octree refinement strategy. Besides, we incorporate the concept of contact potential volume into the subdivided stopping criterion. Compared to the uniform SDF, this approach demonstrates higher generation speed and smaller errors. Several examples are used to verify the accuracy, convergence and efficiency of the proposed formulation. It is observed that the invariance property of SE(3) description leads to enhanced computational performance. Both frame invariance and SDF-based approach are expected to improve efficiency for large-scale problems. The presented method is easily applicable to other types of flexible bodies.

Efficient integration of deep neural networks in sequential multiscale simulations

AbstractMultiscale computations involving finite elements are often unfeasible due to their substantial computational costs arising from numerous microstructure evaluations. This necessitates the utilization of suitable surrogate models, which can be rapidly evaluated. In this work, we apply a purely data‐based deep neural network as a surrogate model for the microstructure evaluation. More precisely, the surrogate model predicts the homogenized stresses, which are typically obtained by the solution of (initial) boundary‐value problems of a microstructure and subsequent homogenization. Furthermore, the required consistent tangent matrix is computed by leveraging reverse mode automatic differentiation. To improve data efficiency and ensure high prediction quality, the well‐known Sobolev training is chosen for creating the surrogate model. This surrogate model is seamlessly integrated into a Fortran‐based finite element code using an open‐source library. As a result, this integration, combined with just‐in‐time compilation, leads to a speed‐up of more than 6000× , as demonstrated in the example of a plate with a hole examined in this work. Furthermore, the surrogate model proves to be applicable even in load‐step size controlled simulations, where it overcomes certain load‐step size limitations associated with the microstructure computations.

An efficient framework for matrix-free SpMV computation on GPU for elastoplastic problems

High computational cost in elastoplastic analysis is often handled by the use of high performance parallel computers. However, the presence of both elastic and plastic states leads to the branching issue, which prevents the realization of true parallel performance. The computational efficiency of an elastoplastic analysis using finite element method is largely determined by the performance of repeated solution of linearized system of equations. In this paper, we propose GPU-based matrix-free strategies to compute sparse matrix–vector multiplication (SpMV) in Conjugate Gradient (CG) iterative solver for the acceleration of solution of linear system of equations. Matrix-free solvers never assemble large sparse global tangent matrix and perform the computation directly with small dense elemental matrices, reducing the storage requirement and preventing the use of problematic sparse storage formats. A uniform treatment of elements in elastic and plastic regions is achieved by the proposed single kernel strategy, which prevents branching, avoids redundant computation and provides efficient memory access. In addition, we propose node-based and degrees-of-freedom (DOF)-based parallel strategies for effective implementation of matrix-free SpMV on a GPU. The proposed strategies use single elemental tangent matrix for all elements in elastic region and individual tangent matrices in plastic region. The computational experiments over three large-scale benchmark examples of elastoplasticity reveal that the performance of the node-based and DOF-based parallelization strategies depends on the amount of plasticity in the body. At low plasticity levels, node-based strategy performs best, achieving 3.2× speedup over an existing GPU-based matrix-free SpMV strategy in the literature. For moderate to high amount of plasticity, the DOF-based strategy outperforms every other strategy and obtains speedups of up to 3.5× over the existing SpMV strategy.

A co-rotational formulation for quasi-steady aerodynamic nonlinear analysis of frame structures

The design of structures submitted to aerodynamic loads usually requires the development of specific computational models considering fluid-structure interactions. Models using structural frame elements are developed in several relevant applications such as, the design of advanced aircraft wings, wind turbine blades or power transmission lines. In the case of flexible frame structures submitted to fluid flows, the computation of inertial and aerodynamic forces for large displacements and rotations is a challenging task. In this article, we present a novel formulation for the efficient computation of aerodynamic forces in frame structures, coupling the co-rotational framework with the quasi-steady theory. A numerical procedure is provided considering a tangent matrix for the aerodynamic forces. This formulation is implemented in the open-source library ONSAS, allowing users to reproduce the results or solve other frame nonlinear dynamic problems. The proposed formulation and its implementation are validated through the resolution of four numerical examples. The formulation and the numerical procedure proposed efficiently provide accurate solutions for these challenging problems with large displacements and rotations.

Strain gradient plasticity with nonlinear evolutionary energetic higher order stresses

This paper explores the effect of evolutionary-type nonlinear energetic higher order stress inside the strain gradient plasticity framework that allows nonlinearity in dissipation. Based on the fundamental principles of thermodynamics, a multi-term Chaboche-type model has been formulated to describe the evolution of energetic moment stresses that depend on strain gradient and effective plastic strain. The derived evolutionary relation shows a coupling between energetic and dissipative length scales. The energetic part of the higher order stress, related to the GND formation, saturates as effective plastic strain increases during plastic flow. Higher order stress update equation and material tangent are derived using implicit time discretization. The proposed nonlinear energetic moment stress model confirms its applicability compared to experimentally observed responses. Our study demonstrates the importance of incorporating a nonlinear kinematic hardening rule to accurately represent the Bauschinger effect in samples with a thickness of less than 1μm. The cyclic bending experiment with a single slip sample shows that the nonlinear kinematic hardening rule can effectively capture nonlinearity in the experimental observation. Compressed shear layer simulation reveals that consideration of thickness-dependent relaxation coefficient accurately models the experimental observations. Thus, it makes further relevant consideration of a more advanced kinematic hardening rule through the proposed approach.

FE2 Computations with Deep Neural Networks: Algorithmic Structure, Data Generation, and Implementation

Multiscale FE2 computations enable the consideration of the micro-mechanical material structure in macroscopical simulations. However, these computations are very time-consuming because of numerous evaluations of a representative volume element, which represents the microstructure. In contrast, neural networks as machine learning methods are very fast to evaluate once they are trained. Even the DNN-FE2 approach is currently a known procedure, where deep neural networks (DNNs) are applied as a surrogate model of the representative volume element. In this contribution, however, a clear description of the algorithmic FE2 structure and the particular integration of deep neural networks are explained in detail. This comprises a suitable training strategy, where particular knowledge of the material behavior is considered to reduce the required amount of training data, a study of the amount of training data required for reliable FE2 simulations with special focus on the errors compared to conventional FE2 simulations, and the implementation aspect to gain considerable speed-up. As it is known, the Sobolev training and automatic differentiation increase data efficiency, prediction accuracy and speed-up in comparison to using two different neural networks for stress and tangent matrix prediction. To gain a significant speed-up of the FE2 computations, an efficient implementation of the trained neural network in a finite element code is provided. This is achieved by drawing on state-of-the-art high-performance computing libraries and just-in-time compilation yielding a maximum speed-up of a factor of more than 5000 compared to a reference FE2 computation. Moreover, the deep neural network surrogate model is able to overcome load-step size limitations of the RVE computations in step-size controlled computations.

Unified three-dimensional finite elements for large strain analysis of compressible and nearly incompressible solids

This work proposes a displacement-based finite element model for large strain analysis of isotropic compressible and nearly-incompressible hyperelastic materials. Constitutive law is written in terms of invariants of the right Cauchy-Green tensor; coupled and decoupled formulations of strain energy functions are presented, whereas a penalty function is used to impose an incompressibility constraint. Based on a total Lagrangian formulation, the nonlinear governing equations are thus obtained by employing the principle of virtual displacements. Analytic expression of both internal forces vector and tangent matrix of linear and high-order hexahedral finite elements are derived by adopting a three-dimensional formalism based on the Carrera Unified Formulation. Popular benchmark problems in hyperelasticity are analyzed to establish the capabilities of the present implementation of fully-nonlinear solid elements in the case of compressible and nearly-incompressible beams, cylindrical shells, and curved structures.

Second-order two-cycle analysis of frames based on interpolation functions from the solution of the beam-column differential equation

In geometrically nonlinear problems solved using the Finite Element Method (FEM), the structure response is directly influenced by the level of discretization and the nonlinear solution algorithm used. To reduce the discretization dependence, exact solutions are developed based on the deformed infinitesimal element equilibrium. To deal with the nonlinear solution problem, the two-cycle method can be used, since it is not dependent on load or displacement steps. The two-cycle method developed by Chen & Lui (1991) uses the classical geometric matrix and is not accurate for high axial loads. This happens because the geometric matrix is obtained using Hermitian polynomials which are approximate solutions. To circumvent this issue, the frame element’s tangent matrix is obtained using interpolation functions that match the homogeneous solution of the differential equation of the beam-column problem. The main objective of this study is to carry out a second order analysis of the frames and obtain equilibrium paths using the two-cycle method and the tangent stiffness matrix based on solutions of the differential equations obtained from the element’s deformed configuration. The results in terms of displacements and rotations for the examples studied are identical to the analytical solutions, showing that the combination of the two-cycle method with the exact element formulation is promising and can diminish the need for discretization.

A kinematically stabilized linear tetrahedral finite element for compressible and nearly incompressible finite elasticity

We propose a stabilized linear tetrahedral finite element method for static, finite elasticity problems involving compressible and nearly incompressible materials. Our approach relies on a mixed formulation, in which the nodal displacement unknown filed is complemented by a nodal Jacobian determinant unknown field. This approach is simple to implement in practical applications (e.g., in commercial software), since it only requires information already available when computing the Newton–Raphson tangent matrix associated with irreducible (i.e., displacement-based) finite element formulations. By nature, the proposed method is easily extensible to nonlinear models involving visco-plastic flow. An extensive suite of numerical tests in two and three dimensions is presented, to demonstrate the performance of the method.