Total Lagrangian Research Articles

An improved total Lagrangian SPH method for modeling solid deformation and damage

Total Lagrangian - Smoothed Particle Hydrodynamics (TL-SPH) is a classical algorithm to improve the tensile instability encountered in the conventional Eulerian kernel-based SPH method for solid mechanics. Meanwhile, the gradient correction usually is employed in TL-SPH to enhance the consistency of approximation, which would generate unavoidable time costs. Therefore, a simplified gradient correction under the initial configuration is derived. Aiming at different particle configurations, the corresponding selection modes are provided and verified. Then the simplified gradient correction is introduced into TL-SPH, which is the Improved TL-SPH (ITL-SPH) method. Firstly, a classical example – elastic rubber collision is simulated to verify ITL-SPH has the same ability as TL-SPH to improve tensile instability. Then, the convergence and high efficiency of ITL-SPH is demonstrated via the uniaxial tension of a plate. Moreover, the plate with square and triangle holes also is simulated to show the adaptation and stability of ITL-SPH for the extreme boundary. Under the Total Lagrangian frame, the treatment to model the crack is necessary. So, this paper proposes a damage model and combines it with the ITL-SPH method for crack initiation and propagation, which is verified by the plate with holes and the notched beam examples.

NUMERICAL SIMULATION FOR FLUID-STRUCTURE INTERACTION OF A BLOOD FLOW WITH THE AORTIC VALVE USING THE FEM MONOLITHIC FORMULATION

The paper aims to present a numerical simulation for fluid-structure interaction (FSI) of a blood flow over the aortic valve. The finite element discretization is adopted both for fluid and solid domains. The monolithic scheme is used for the strong coupling of fluid and structure to satisfy kinematic and dynamic equilibrium conditions at the interface. The Navier-Stokes equations of an incompressible flow are solved by using the integrated method based on the ALE formula for the moving grid, and the total Lagrangian formulation is used for the non-linear hyper-elastic material of the aortic valve with the Mooney-Rivlin material model adopted as a constitutive equation for the solid domain. The monolithic FEM method is validated by solving a 3D pressure wave problem and the results are compared to the previous solutions. And then, the present method is employed to investigate blood flow through the aortic valve with a complex geometry. The simulation results can be used for predicting the risk of aortic valve diseases…

On material selection for topology optimized compliant mechanisms

Although compliant mechanism design is a thoroughly studied field, surprisingly little information can be found in literature regarding selection of optimal materials. This paper is intended to fill this gap. Density-based, geometrically robust, stress constrained topology optimization based on a total Lagrangian FEM formulation is used for investigation of a compliant inverter and a compliant gripper example. Changes in handling of the projection parameters and the stress constraints are proposed for improved algorithmic stability and accurate representation of the stresses and material stiffness in topology optimization. Large-scale optimization studies are carried out, varying elastic modulus and allowable stress in a wide range. A comparison with an Ashby chart shows the characteristics of the best suited material. It is shown, that an optimal Young’s modulus and a minimum required material strength (depending on the modulus) can be identified from the results. Altering critical optimization parameters, e.g. allowable volume fraction and the minimum length scale, their influence on the optimal material choice is investigated. Guidelines for compliant mechanism designers for efficient selection of suited materials are developed.

Nonlinear dynamic analysis of FG carbon nanotube/epoxy nanocomposite cylinder with large strains assuming particle/matrix interphase using MLPG method

In this paper, a first known study on the large deformation dynamic behavior of functionally graded carbon nanotube-reinforced (CNTR) nanocomposite cylinder considering the particle/matrix interphase property is presented. The geometrically nonlinear dynamic analysis is carried out using the meshless local Petrov–Galerkin (MLPG) method based on the total Lagrangian approach. The obtained nonlinear time dependent equations are solved using the incremental-iterative Newmark/Newton–Raphson technique. The effective properties of CNTR cylinder is evaluated by employing the three-phase Halpin–Tsai model and rule of mixture. In these models, three constituent phases including nanoparticle, interphase and matrix is considered. It is assumed that CNTs is dispersed in the resin matrix with four different function along the radial direction to obtain the optimum grading pattern. The effects of interphase properties, CNT's weight fraction, CNT distribution pattern and volume fraction exponent on the dynamic characteristics of the cylinder are discussed in details. Numerical results demonstrated the high performance of the proposed MLPG method for geometrically nonlinear dynamic analysis of functionally graded carbon nanotube/epoxy nanocomposites due to its advantage in eliminating the mesh distortion and high capability in FG material modeling. Moreover, it is found that the interphase properties may dramatically affect the dynamic behavior of nanocomposite cylinder especially at the higher CNT volume fractions. So, it is important to consider the interphase properties to correctly model the nanocomposite structures.

Effect of different geometrically nonlinear strain measures on the static nonlinear response of isotropic and composite shells with constant curvature

The structural analysis of ultra-lightweight flexible shells and membranes may require the adoption of complex nonlinear strain-displacement relations. These may be approximated and simplified in some circumstances, e.g., in the case of moderately large displacements and rotations, in some others may be not. In this paper, the effectiveness of various geometrically nonlinear strain approximations such as the von Krmn strains is investigated by making use of refined shell formulations based on the Carrera Unified Formulation (CUF). Furthermore, geometrical nonlinear equations are written in a total Lagrangian framework and solved with an opportune Newton-Raphson method. Test cases include the study of shells subjected to pinched loadings, combined flexure and compression, and post-buckling including snap-through problems. It is demonstrated that full geometrically nonlinear analysis accounting for full Green-Lagrange strains shall be performed whenever displacements are higher than the order of magnitude of the thickness and if compressive loads are applied.

A Mindlin shell model based on the corrective smoothed particle method and accuracy implementation of the free boundary

A meshless shell method for large deformation and a linear relation between the Cauchy stress tensor and the Almansi strain tensor based on the corrective smoothed particle method (CSPM) is presented in this paper. Due to the use of the shell theory, only one layer of particles in the reference plane is required to discretize the shell model. The CSPM combining the kernel estimate with the Taylor series expansion is adopted, which resolves the general problems of low precision and particle deficiency in standard smoothed particle hydrodynamics (SPH). The discrete governing equations of the shell in the strong form are derived using the conservation condition and the CSPM interpolation function. Aiming at the sore point of the free boundary in the meshless method, the developed model enables the modified governing equations to automatically satisfy the free boundary condition without additional treatment. Moreover, the total Lagrangian kernel function and stress points are employed to eliminate tensile instability and instability induced by the rank deficiency. Finally, several numerical examples are used to verify the validity and accuracy of the meshless shell model.

A direct Jacobian total Lagrangian explicit dynamics finite element algorithm for real‐time simulation of hyperelastic materials

AbstractThis article presents a novel direct Jacobian total Lagrangian explicit dynamics (DJ‐TLED) finite element algorithm for real‐time nonlinear mechanics simulation. The nodal force contributions are expressed using only the Jacobian operator, instead of the deformation gradient tensor and finite deformation tensor, for fewer computational operations at run‐time. Owing to this proposed Jacobian formulation, novel expressions are developed for strain invariants and constant components, which are also based on the Jacobian operator. Results show that the proposed DJ‐TLED consumed between 0.70× and 0.88× CPU solution times compared to state‐of‐the‐art TLED and achieved up to 121.72× and 94.26× speed improvements in tetrahedral and hexahedral meshes, respectively, using GPU acceleration. Compared to TLED, the most notable difference is that the notions of stress and strain are not explicitly visible in the proposed DJ‐TLED but embedded implicitly in the formulation of nodal forces. Such a force formulation can be beneficial for fast deformation computation and can be particularly useful if the displacement field is of primary interest, which is demonstrated using a neurosurgical simulation of brain deformations for image‐guided neurosurgery. The present work contributes towards a comprehensive DJ‐TLED algorithm concerning isotropic and anisotropic hyperelastic constitutive models and GPU implementation.

Oceanic versus terrestrial origin of El Niño Southern Oscillation-associated continental precipitation anomalies.

Precipitation is significantly influenced by the El Niño Southern Oscillation (ENSO), which is considered to be the most important factor that brings about climate variability. In this study, the asymmetry of the origin of continental precipitation anomalies during El Niño and La Niña events was investigated. An already validated Lagrangian approach was used to determine the proportion of the total Lagrangian precipitation that is of oceanic and terrestrial origin. Further, the role of these components of the Lagrangian precipitation in regions with significant precipitation anomalies during the ENSO was investigated. A two-phase asymmetric behavior of precipitation, particularly in tropical regions, was obtained. For some of these regions, precipitation anomalies based on other datasets were also calculated to confirm the observed changes, and for these regions, it was observed that in all cases, the calculated anomaly of Lagrangian precipitation agreed with the precipitation change. The percentage of precipitation of oceanic origin was higher in most of these regions. However, it was observed that an increase in the percentage of precipitation of oceanic origin did not bring about a general increase in precipitation for all the regions, revealing the importance of precipitation of terrestrial origin during both phases of the ENSO.

Dynamic analysis of flat and folded laminated composite plates under hygrothermal environment using a nonpolynomial shear deformation theory

In this paper, the effect of the hygrothermal environment on the dynamic analysis of one-fold and two-fold folded laminated composite plates using a nonpolynomial shear deformation theory (NPSDT) is investigated. A computationally efficient C0 finite element method (FEM) is applied to examine the free vibration characteristic, transient behavior, and steady-state response of laminated composite plates. The formulation employs the total Lagrangian approach for analysis, and the initial stress generated due to hygrothermal load is taken into account through the Green–Lagrange strain displacement relation. The Newmark’s method is employed to integrate the spatial–temporal partial differential governing equations. The effect of the thermal environment on the natural frequency of folded and flat composite plates has been illustrated through various examples. The transient displacement and stresses are plotted for various loads. Moreover, transient damped analysis under the thermal load is carried out by considering the Rayleigh damping model. Further, linear harmonic analysis under the hygrothermal environment has also been carried out, and the responses in the neighborhood of natural frequency are plotted. The present model has been compared and validated by the existing literature and the obtained ANSYS APDL solutions, and is found to be performing better for NPSDT.

Second-Order Analytic Derivatives for XYG3 Type of Doubly Hybrid Density Functionals: Theory, Implementation, and Application to Harmonic and Anharmonic Vibrational Frequency Calculations.

Analytic derivative methods in quantum chemistry are powerful tools for the calculation of molecular properties and simulation of chemical systems. While the derivatives of the well-established B2PLYP type of doubly hybrid (DH) density functionals can be generated by a straightforward combination between the Kohn-Sham density functional and the second-order perturbation theory (PT2), both of these two contributions have to be considered nonvariationally for the XYG3 type of DH functionals (xDHs). A total Lagrangian that includes both parts is therefore needed for the corresponding Z-vector equations for the first-order derivatives of xDHs. Starting from the differentiation of the Z-vector equations, a theory for the second-order derivatives for xDHs is developed here and is applied to the molecular harmonic and anharmonic vibrational frequency calculations. The results are generally of high quality, as compared to the well-established experimental and CCSD(T) counterparts. Further investigations on the fundamental frequency predictions prove the capability of the xDH functionals for an accurate calculation of spectroscopic properties for a wide range of medium-size molecules.

Isogeometric analysis for buckling and postbuckling of graphene platelet reinforced composite plates in thermal environments

This work aims at examining the buckling and postbuckling behaviour of laminated composite plate in which GPLs (graphene platelets) are functionally dispersed under various kinds of thermal loadings. For this purpose, IGA (isogeometric analysis) method of which merits encompass the precise geometric representation, smoothness of higher-order, cost-efficiency and high accuracy is taken advantage of. Shear deformable plate theory into which a hyperbolic shear shape function and the von Kármán type nonlinear kinematics are incorporated is used to derive the nonlinear equations of equilibrium for the GPL reinforced composite plate subjected to thermal gradients. On the basis of the total Lagrangian formulation, the governing equation for the thermal buckling is established and the nonlinear thermal postbuckling path is traced together with the modified Newton-Raphson iteration. Three schemes of the symmetric GPL distributions are considered. The IGA approach proposed here, by checking the performance capacity via the multiple numerical tests, is substantiated to successfully follow the nonlinear buckling and postbuckling path of the nanocomposite plate in thermal environments. The further parametric investigations highlight the impacts of the thermal load type, GPL weight fraction, GPL arrangement pattern, plate geometric parameter, plate constraint condition and surface temperature contour on the thermal buckling and postbuckling responses of the GPL strengthened composite plate.

Post-buckling and large-deflection analysis of a sandwich FG plate with FG porous core using Carrera’s Unified Formulation

In this study, the unified formulation of a full geometrically nonlinear refined plate theory in a total Lagrangian approach is developed to study the post-buckling and large-deflection analysis of sandwich functionally graded (FG) plate with FG porous (FGP) core. The plate has three layers so that the upper and lower layers are FG and the middle layer (core) is the FGP, which is considered with four cases in terms of the porosity core distribution. The different two-dimensional (2D) plate structures kinematics are consistently implemented based on the Carrera’s Unified Formulation (CUF) by means of an index notation and an arbitrary expansion function of the generalized variables in the thickness direction, leading to lower- to higher-order plate models with only pure displacement variables. Furthermore, a finite element approximation and the principle of virtual work are used to easily and straightforwardly formulate the nonlinear governing equations in a total Lagrangian manner, whereas a path-following Newton-Raphson linearization scheme based on the arc-length constraint is utilized to solve the full geometrically nonlinear problem. Numerical assessments are finally conducted to confirm the capabilities of the proposed CUF plate model to predict the post-buckling and large-deflection equilibrium curves with high accuracy.

Three-dimensional nonlinear mixed 6-DOF beam element for thin-walled members

This paper presents a three-dimensional mixed beam element formulation for fully nonlinear distributed plasticity analysis of members composed of sections with no significant torsional warping such as steel angles and tees. This formulation is presented using a corotational total Lagrangian approach and implemented in the OpenSees corotational framework. In this context, a basic coordinate system is lined up with the element chord and translates and rotates as the element deforms. The element tangent stiffness matrix and resisting forces in the basic system are derived through linearization of the two-field Hellinger-Reissner variational principle. The displacement shape functions are cubic Hermitian functions for the transverse displacements and a linear shape function for the axial and torsional deformation. The generalized stress resultant shape functions are linear for moments and constant for axial force and torque with the P - δ effect considered, which are developed from equilibrium equations. The fiber section method with uniaxial constitutive laws is adopted to account for material nonlinearity. Since the degrees-of-freedom in the basic system are defined with respect to different reference points, all element responses are transformed to acting about the shear center before conducting the corotational transformation. The mixed element is validated through a number of experimental and numerical examples.

Three-dimensional nonlinear displacement-based beam element for members with angle and tee sections

Asymmetric thin-walled sections such as steel angles and tees are widely used in a range of steel structures. To address extreme limit states that these structures encounter due to extreme events such as hurricanes and earthquakes, it is important to capture their response due to large deformations caused by static or dynamic loading. In the nonlinear large deformation regime, these members have coupled axial-flexural–torsional deformation due to the so-called Wagner effect and the noncoincident shear center and centroid. A three-dimensional corotational total Lagrangian beam element is formulated and implemented in the OpenSees corotational framework to account for these coupling effects by invoking Green-Lagrange strains referenced to a basic system. In the basic system, shear forces and torque are defined with respect to the shear center, axial force is referred to the centroid, and flexure is defined around the section principle axes but in the planes containing the shear center. The element tangent stiffness matrix is derived through linearization of the governing equation obtained from the principle of virtual work. Cubic Hermitian functions for the transverse displacements and a linear shape function for the axial and torsional deformation are adopted in the development. Before conducting the corotational transformation, all element end forces and displacements are transformed to act about the shear center. In order to remedy membrane locking in the inextensional bending mode, the high order bending terms in the axial strain are replaced by a constant effective strain. Cyclic material nonlinearity is considered by discretizing the cross section into a grid of fibers, tracking the steel uniaxial stress–strain constitutive at each fiber, and performing numerical integration over the cross section to obtain the section stiffness matrix. The formulation is compared against a set of experimental and numerical results to validate that the element can model geometric and material nonlinearities accurately and efficiently.

Stress States in Highly Flexible Thin-Walled Composite Structures by Unified Shell Model

This work deals with highly flexible laminated shell structures. The main focus is to provide an advanced model for the accurate prediction of the interlaminar three-dimensional stress state of shells subjected to large displacements/rotations, buckling, and snap-through phenomena. In this context, a two-dimensional shell finite element based on the Carrera unified formulation (CUF) is formulated in an orthogonal curvilinear reference system. Thanks to CUF, the governing equations are expressed in terms of fundamental nuclei, which are invariant of the shell theory approximation order. Thus, classical theories of structures to layerwise approaches can be implemented with ease and in a unified manner. The full Green–Lagrange strain tensor is employed because far nonlinear regimes are investigated. Furthermore, the geometrical nonlinear equations are written in a total Lagrangian framework and solved with an opportune Newton–Raphson method along with a path-following approach based on the arc-length constraint. The results demonstrate that classical theories may bring to wrong and unconservative stress predictions, especially in nonlinear equilibrium states, where the use of advanced and layerwise approaches shall be recommended.

Coupling total Lagrangian SPH–EISPH for fluid–structure interaction with large deformed hyperelastic solid bodies

In this work, we propose a two-way coupling technique between a total Lagrangian smoothed particle hydrodynamics (SPH) method for Solid Mechanics and the explicit incompressible SPH (EISPH) to simulate fluid–structure interaction problems. In the solid part, the total Lagrangian framework guarantees that the particle distribution keep stable to correctly calculate the deformation gradient and thus the elastic forces. The constitutive model follows hyperelastic formulations, and the stability of the method is enforced by a Jameson–Schmidt–Turkel (JST) stabilization procedure. For the fluid part, we applied an EISPH formulation, which is a fully explicit incompressible scheme based on a projection method capable of providing accurate pressure distributions for free-surface flows, while avoiding costly linear equations. The coupling scheme follows the same manner as the fixed wall ghost particle (FWGP) approach, which was here adapted to include moving walls. In addition, the non-penetration condition is rigorously reinforced through a numerical algorithm to avoid penetration of every fluid particle, including free-surface particles. Our method for solid is then verified through a large deformed tension plate numerical test, and our coupling forces through a series floating tests and hydrostatic water column over a thin infinite plate. Then, the method is validated comparing it with experimental data of a dam break test in which the water column attacks a thin rubber plate.

A higher-order quadrilateral shell finite element for geometrically nonlinear analysis

In this paper, we present a geometrically nonlinear formulation for a nine-node shell finite element and demonstrate its performance. The total Lagrangian formulation is utilized to allow for large displacements and large rotations. The mixed interpolation of tensorial components (MITC) technique is applied to effectively reduce the membrane and shear locking phenomena without refining mesh, introducing additional nodes, and using reduced integrations. Many-core accelerators with GPU-compatible libraries are used to achieve the highest speed-up for the solution process. The performance of the present nine-node shell element is compared with those of the four-node and six-node shell elements in several benchmark problems, including Cook's skew beam, cantilever plate, pinched cylindrical shell, slit annular plate, and hemisphere shell. The present element achieves excellent performance even when the coarse mesh is used.

A Numerical Framework for Geometrically Nonlinear Deformation of Flexoelectric Solids Immersed in an Electrostatic Medium

Abstract A numerical formulation coupling finite and boundary element methods is developed to analyze the mechanical deformation and electric polarization of flexoelectric solids experiencing geometrically nonlinear deformation. The proposed method considers the electrical interactions among flexoelectric solids, electric charges, and their surrounding medium. First, a higher-order gradient theory is proposed based on the skew-symmetric couple-stress model to analyze the geometrically nonlinear deformation of flexoelectric solids. This theory includes a total Lagrangian weak form that satisfies linear momentum conservation, angular momentum conservation, and Gauss’s law. Based on the proposed theory, a finite element is developed using basis functions that satisfy C1 continuity. Second, a coupled formulation is developed to consider the electrical interactions among solids, electric charges, and their surrounding medium. In this formulation, conventional boundary elements are adopted to account for the electrostatic surroundings. Besides, electric boundary conditions are naturally imposed on solid boundaries according to the electrical interactions between solids and their electrostatic surroundings. Finally, the proposed method is validated via the comparisons of the numerical results with closed-form solutions.

A nonlinear plane strain finite element analysis for multilayer elastomeric bearings

The purpose of this paper is the prediction of the behaviour of multilayer elastomeric bearings by the finite element analysis. First, the finite element formulation of equilibrium equations of incompressible elasticity problem is presented. This approach is based on the modified potential energy taking into account the incompressibility. The linearized system obtained is solved by the Newton–Raphson iterative method with the total lagrangian formulation. A large plane strain element is developed for the analysis of elastomeric materials under large strain deformation. This element is then integrated into our software. After validating the plane strain element using examples with known exact solutions, an industrial case study made of laminated (metal-rubber) elastomeric component is examined. The stress distribution study is limited to the elastomeric layers, possible sources of crack appearance.

Dynamic analysis of failure paths of truss structures: Benchmark examples including material degradation

We propose novel benchmark examples of dynamic failure paths of truss structures. Novelty arises from use of a logarithmic strain measure, within a total Lagrangian formulation, combined with a continuum material damage model. Previous benchmarks have considered the usual engineering strain measure, which is not always useful, as it can lead to material degeneration under finite levels of stress. Ductile material behavior has been considered in the literature, addressing plasticity and some material softening, but neglecting material degradation. Herein, damage accumulation is associated to the hydrostatic component of plastic strains, leading to a stable and explicit representation of material degradation. The original static formulation, presented elsewhere by the authors, is extended herein to the dynamic analysis of failure paths of truss structures. Several numerical examples from the literature are studied, and the differences in dynamic behavior are pointed out. In our implementation, as individual bars are damaged, elastic unloading and load redistribution are observed. When the critical damage hypothesized by Lemaitre is reached by individual bars, these are fully unloaded, with no material degeneration or numerical instability. Numerical results highlight the significant effects of material degradation in the dynamic behavior of truss structures under exceptional loads. We propose the set of examples addressed herein as the new benchmarks to which future developments in geometrical and material non-linear truss modelling will be compared.