Total Lagrangian Research Articles

An improved radial basis reproducing kernel particle method for geometrically nonlinear problem analysis of SMAs

In this paper, the radial basis function (RBF) without shaped parameter is utilized in the radial basis reproducing kernel particle method (RRKPM), and an improved radial basis reproducing kernel particle method (IRRKPM) is proposed. Compared with traditional RKPM, the IRRKPM effectively reduces the impact of different kernel functions on calculation precision, and is further employed to examine geometrically nonlinear problems associated with shape memory alloys (SMAs). The displacement boundary condition is enforced via the penalty function method, while the Galerkin integration method in its weak form, along with the total Lagrangian (TL) approach, is utilized to derive the geometrically nonlinear equations for SMAs within the IRRKPM framework. The equilibrium equations are then solved using the Newton Raphson (N-R) iterative method. The impact of the different penalty factor and the radius control parameter of influence domain on errors is analyzed, the computational precision of the IRRKPM is compared with the RRKPM, and the computational stability is evaluated. Finally, the suitability of the IRRKPM for the analysis of geometrically nonlinearity problems in SMAs are confirmed through specific numerical examples.

Unified non-hourglass formulation for total Lagrangian SPH solid dynamics

AbstractThe persistence of hourglass modes poses a significant numerical instability issue in total Lagrangian smoothed particle hydrodynamics (TLSPH) solid dynamics, especially when dealing with substantial deformations, regardless of material properties. However, existing hourglass control methods have shown effectiveness only within limited applications. Thus far, a comprehensive solution capable of addressing hourglass issues across a wide range of material models, including elasticity, plasticity, and anisotropy, remains elusive. In this study, we introduce a unified TLSPH formulation grounded in volumetric-deviatoric stress decomposition, aimed at fundamentally mitigating hourglass modes in general simulations. Different conceptually from previous approaches using stress points or extra viscous or hourglass-control stresses within the momentum equation, our formulation is based on the weighted average of a standard but hourglass-prone formulation and an essentially non-hourglass formulation for elastic materials, employing a single limiter to dynamically adjust the weighting between the two formulations. Crucially, the dimensionless characteristic of the formulation enables seamless handling of complex material models. To validate the effectiveness of our formulation, we conduct simulations across a range of benchmark cases involving elastic, plastic, and anisotropic materials. To illustrate its versatility, we apply the formulation to simulate a complex scenario involving viscous plastic Oobleck material, contacts, and very large deformation. Our work addresses a critical gap in TLSPH simulations by offering a unified approach to mitigate hourglass modes, enhancing the reliability and accuracy of simulations across diverse material models and complex scenarios.

Nonlinear analysis of spatial trusses with different strain measures and compressible solid

This paper investigates the nonlinear behavior of spatial truss elements under finite deformations, focusing on the impact of various strain measures in compressible materials. We examine both Total Lagrangian (using engineering and Green–Lagrange strains) and Eulerian formulations (using natural, Biot, and Almansi strains). The analysis assumes a linear spatial hyperelastic material where Cauchy stress is proportional to axial natural strain via Young’s modulus. For infinitesimal strains, Young’s modulus remains consistent across different stress/strain pairs. In the finite strain regime, we derive a nonlinear secant modulus based on Young’s modulus. Internal force vectors and tangent stiffness matrices are computed using the direction cosines of the truss element in its deformed state. The paper demonstrates that for infinitesimal deformations, adjusting the modulus of elasticity when using different stress/strain pairs is unnecessary. However, for finite deformations, it is essential to adjust the modulus of elasticity. Numerical simulations validate the performance of the proposed 3D truss element against established formulations. This research offers critical insights into the nonlinear response of spatial trusses, guiding the selection of appropriate strain measures for enhanced accuracy in engineering applications. These findings contribute to more reliable and efficient structural designs, especially in scenarios involving finite deformations and compressible materials.

NURBS-based isogeometric formulation for linear and nonlinear buckling analysis of laminated composite plates using constrained and unconstrained TSDTs

Two isogeometric plate models employing Reddy's third-order shear deformation theory (TSDT) and unconstrained third-order shear deformation theory (UTSDT) are presented and compared for linear and nonlinear buckling analysis of laminated composite plates with and without imperfection and subjected to different inplane loads. Cubic non-uniform rational B-spline (NURBS) basis functions that easily satisfy C1 continuity of the IGA-TSDT model are employed. The total Lagrangian approach in conjunction with the principle of virtual work is used to derive the governing equations. The primary and secondary solutions are traced using a tangent based arc-length method with a simple branch switching technique. The performance of the models is evaluated by validation and comparison with solutions obtained using ANSYS, Navier method (for linear analysis only), and those in the literature. The buckling response is significantly affected by pre-buckling boundary conditions and strain-displacement relations. The nonlinear buckling approach, among other approaches, is observed to be the most accurate methodology for an arbitrarily laminated composite plate. Further, IGA-UTSDT with nine DOF gives marginal improvement over IGA-TSDT with five DOF at the cost of computation. The IGA-TSDT is observed to be superior to FEM-TSDT in terms of computation demand and performance.

Nonlinear thermo-electro-mechanical responses and active control of functionally graded piezoelectric plates subjected to strong electric fields

Functionally graded piezoelectric plates (FGPPs) are often exposed to extreme thermal environments and subjected to large amplitude excitation and strong electric fields. Accurately predicting the nonlinear behaviors of FGPPs under large thermo-electro-mechanical loads remains a considerable challenge. This paper proposes a comprehensive nonlinear coupled analysis model that considers geometric nonlinearity, piezoelectric nonlinearity, and the temperature dependence of piezoelectric parameters. The total Lagrangian incremental finite element equations for FGPPs under thermo-electro-mechanical loads are derived using Hamilton's principle, by employing the first-order shear deformation theory and the von Kármán nonlinear strain-displacement relationship. The accuracy of the model is validated through comparative studies. Based on this multi-nonlinear model, we further investigate the effects of geometric nonlinearity, piezoelectric nonlinearity, and temperature dependence on the nonlinear static bending, dynamic responses, and active control of FGPPs. This study demonstrates that considering these factors enables accurate prediction of the static, dynamic, and active control behaviors of FGPPs.

HLFEMP: A coupled MPM-FEM method under a hybrid updated and total Lagrangian framework

Coupling the Material Point Method (MPM) with the Finite Element Method (FEM) provides a powerful approach for addressing problems characterized by the coexistence of large and small deformations. The majority of existing MPM-FEM methods, which operate within the updated Lagrangian framework, introduce inherent issues, such as cell-crossing noise and numerical fracture, compromising the overall accuracy of the coupled method. This paper presents a coupled MPM-FEM method under a hybrid updated and total Lagrangian framework (HLFEMP). Within the HLFEMP, updated Lagrangian FEM is employed to handle small deformations, while total Lagrangian MPM is adopted to model large deformations. A particle-to-surface contact algorithm with auxiliary thickness layer is utilized to couple both methods. Notably, by imposing the weak form of total Lagrangian MPM in the undeformed configuration, HLFEMP is rendered free from cell-crossing noise and numerical fracture, resulting in better computational stability. The performance of HLFEMP in terms of energy behavior, accuracy, efficiency and stability under large elastic and elasto-plastic deformations are investigated through a series of numerical simulations.

Nonlinear thermal-mechanical coupled isogeometric analysis for GPLs reinforced functionally graded porous plates

This work presents an isogeometric model for thermal-mechanical coupled analysis of geometrically nonlinear responses of graphene platelets reinforced functionally graded porous (GPLs-FGP) plates. The closed-cell Gaussian random field (GRF) scheme, Halpin-Tsai micromechanics model and rule of mixture are utilized to determine the material properties of GPLs-FGP plates with three types of porosity distribution and GPLs dispersion pattern. Unlike previous studies, the influences of GPLs and porosity on the thermal conductivity of GPLs-FGP plates are considered simultaneously. A geometrically nonlinear isogeometric analysis (IGA) model is developed base on a four-variable refined plate theory (RPT), using the von Kármán nonlinear theory, Hamilton’s principle and Total Lagrangian (TL) incremental scheme. After ensuring the accuracy and reliability of the present model through comparison with several examples, the thermal conductivity, temperature distribution, geometrically nonlinear static bending and transient responses of the plate, taking into account the temperature effects of porosity and graphene under thermal-mechanical loads are investigated. Numerical results show that porosity exerts influences on the thermal conductivity and temperature distribution of GPLs-FGP plates, which subsequently impact their geometrically nonlinear thermal-mechanical coupled responses. The results of this study are helpful for designing and manufacturing GPLs-FGP structures with improved performance.

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.

Simulating plate and shell structures with anisotropic resolution using adaptive smoothed particle hydrodynamics

When simulating plate and shell structures characterized by large aspect ratios, reduced-dimensional models are frequently employed due to their notable reduction in computational overhead in contrast to traditional isotropic full-dimensional models. However, in scenarios involving variations in the thickness direction, where adequate resolution in this dimension is required, reduced-dimensional models exhibit limitations. To capture variations in the thickness direction while simultaneously mitigating computational costs, an anisotropic full-dimensional model, integrated with an adaptive smoothed particle hydrodynamics method (ASPH), is developed for simulating behaviors of plate and shell structures in this study. The correction matrix, which is applied to ensure the first-order consistency, is modified accordingly by incorporating the nonisotropic kernel into it within the total Lagrangian framework of ASPH. A series of numerical examples, along with a specific application concerning the deformation of a porous film due to nonuniform internal fluid pressure in the thickness direction, are conducted to assess the computational accuracy and efficiency of the proposed ASPH method. Comparative analyses of our results against reference data and traditional isotropic SPH solutions demonstrate close agreements, affirming the suitability of the present ASPH method across various scenarios.

An efficient truncation scheme for Eulerian and total Lagrangian smoothed particle hydrodynamics methods

In smoothed particle hydrodynamics (SPH) method, the particle-based approximations are implemented via kernel functions, and the evaluation of performance involves two key criteria: numerical accuracy and computational efficiency. In the SPH community, the Wendland kernel reigns as the prevailing choice due to its commendable accuracy and reasonable computational efficiency. Nevertheless, there exists an urgent need to enhance computational efficiency while upholding accuracy. In this paper, we employ a truncation approach to limit the compact support of the Wendland kernel to 1.6h. This decision is based on the observation that particles within the range of 1.6h to 2h make negligible contributions to the SPH approximation. To decrease numerical errors from SPH approximation and the truncation method, we incorporate the Laguerre–Gauss kernel for particle relaxation to obtain the high-quality particle distribution with reduced residue [Wang et al., “A fourth-order kernel for improving numerical accuracy and stability in Eulerian and total Lagrangian SPH,” arXiv:2309.01581 (2023)], and the kernel gradient correction to rectify integration errors. A comprehensive set of numerical examples including fluid dynamics in Eulerian formulation and solid dynamics in total Lagrangian formulation are tested and have demonstrated that truncated and non-truncated Wendland kernels enable achieving the same level of accuracy but the former significantly increases the computational efficiency.

Total Lagrangian smoothed particle hydrodynamics with an improved bond-based deformation gradient for large strain solid dynamics

Total Lagrangian Smoothed Particle Hydrodynamics (TLSPH) is one variant of SPH where the variables are described using the fixed reference configuration and a Lagrangian smoothing kernel. TLSPH elevates the computational efficiency of the standard SPH when no topological change is involved, and it alleviates the stability of SPH scheme with respect to tensile loading. However, instabilities associated with spurious mode, or hourglass/zero-energy mode, persists and often affects the simulation of solids undergoing extremely large strain. This work proposes an alternative formulation to compute deformation gradient with improved accuracy and therefore minimising the possibility of encountering the zero-energy mode. Specifically, we leverage the local discrete computation of bond-based (or pairwise) deformation gradient smoothed by the kernel. Additionally, the bond of a particle with itself is considered to preserve the polynomial reproducibility imposed by the kernel correction scheme. We showcase the convergence of the approach using a two-dimensional benchmark example. Furthermore, the accuracy, robustness, and stability of the proposed approach are assessed in various two- and three-dimensional examples, highlighting on the stability improvement that allows for solid dynamic simulations with more extreme elongation.

Geometrically nonlinear transient analyses of rotating structures through high-fidelity models

This work presents geometrically nonlinear transient analyses of various rotating blades. The structures are discretized through refined beams or multi-dimensional finite element models, which are generated using the Carrera Unified Formulation (CUF). The CUF offers a procedure to develop low- and high-fidelity one-dimensional, two-dimensional, and three-dimensional finite element models hierarchically and automatically. Various beam models were developed using different kinematics models based on Taylor or Lagrange expansion functions. Multi-dimensional models were obtained by merging beam and solid elements, exploiting the unique feature of Lagrange polynomials to have only pure displacements as unknowns. This property allows beam and solid elements to be coupled at the node level without requiring complicated mathematical formulations. By utilizing the Finite Element Method in conjunction with the CUF, the governing equations are written by including all rotation effects, namely the Coriolis term, spin-softening, and geometrical stiffening. In a total Lagrangian scenario, the Hilbert-Hughes-Taylor-α method and the iterative Newton-Raphson scheme are employed to solve the equations of motion. The proposed methodology has been applied to evaluate different blade configurations, comparing the solution obtained using linear, linearized, and nonlinear approaches. The results have been verified and validated by comparing them with existing solutions present in the literature.

Finite element approximation of stabilized mixed models in finite strain hyperelasticity involving displacements and stresses and/or pressure—An overview of alternatives

AbstractThis paper presents mixed finite element formulations to approximate the hyperelasticity problem using as unknowns the displacements and either stresses or pressure or both. These mixed formulations require either finite element spaces for the unknowns that satisfy the proper inf‐sup conditions to guarantee stability or to employ stabilized finite element formulations that provide freedom for the choice of the interpolating spaces. The latter approach is followed in this work, using the Variational Multiscale concept to derive these formulations. Regarding the tackling of the geometry, we consider both infinitesimal and finite strain problems, considering for the latter both an updated Lagrangian and a total Lagrangian description of the governing equations. The combination of the different geometrical descriptions and the mixed formulations employed provides a good number of alternatives that are all reviewed in this paper.

Thermomechanical modeling in elastic body with corotated total Lagrangian SPH

In this paper, we propose a novel thermal deformation formulation for elastic bodies' thermomechanical simulation with total Lagrangian smoothed particle hydrodynamics (TLSPH), which employing fixed reference configuration. We utilize implicit corotated TLSPH as a base methodology to address tensile instability and enhance the robustness of simulation. The reference configuration of heat transfer model is updated, but the reference configuration of thermal deformation model is fixed like TLSPH. As particle's temperature is changed, the inter-particle distance in reference configuration is updated by using linear thermal expansion theory. Validation involved two basic cantilever beam loading cases and a thermomechanical analysis with heat-induced thermal deformation on a cantilever beam. The TLSPH simulation results were compared against Finite Element Method simulations and analytical solutions, demonstrating equivalent accuracy across all tests. This study indicates the potential for conducting a wide array of simulations involving thermal deformation using the TLSPH method, which efficiently supports parallel computing on an individual particle basis.

Accelerating aeroelastic UVLM simulations by inexact Newton algorithms

AbstractWe consider the aeroelastic simulation of flexible mechanical structures submerged in subsonic fluid flows at low Mach numbers. The nonlinear kinematics of flexible bodies are described in the total Lagrangian formulation and discretized by finite elements. The aerodynamic loads are computed using the unsteady vortex-lattice method wherein a free wake is tracked over time. Each implicit time step in the dynamic simulation then requires solving a nonlinear equation system in the structural variables with additional aerodynamic load terms. Our focus here is on the efficient numerical solution of this system by accelerating the Newton algorithm. The particular structure of the aeroelastic nonlinear system suggests the structural derivative as an approximation to the full derivative in the linear Newton system. We investigate and compare two promising algorithms based on this approximation, a quasi-Newton type algorithm and a novel inexact Newton algorithm. Numerical experiments are performed on a flexible plate and on a wind turbine. Our computational results show that the approximation can indeed accelerate the Newton algorithm substantially. Surprisingly, the theoretically preferable inexact Newton algorithm is much slower than the quasi-Newton algorithm, which motivates further research to speed up derivative evaluations.

A first‐order hyperbolic arbitrary Lagrangian Eulerian conservation formulation for non‐linear solid dynamics

SummaryThe paper introduces a computational framework using a novel Arbitrary Lagrangian Eulerian (ALE) formalism in the form of a system of first‐order conservation laws. In addition to the usual material and spatial configurations, an additional referential (intrinsic) configuration is introduced in order to disassociate material particles from mesh positions. Using isothermal hyperelasticity as a starting point, mass, linear momentum and total energy conservation equations are written and solved with respect to the reference configuration. In addition, with the purpose of guaranteeing equal order of convergence of strains/stresses and velocities/displacements, the computation of the standard deformation gradient tensor (measured from material to spatial configuration) is obtained via its multiplicative decomposition into two auxiliary deformation gradient tensors, both computed via additional first‐order conservation laws. Crucially, the new ALE conservative formulation will be shown to degenerate elegantly into alternative mixed systems of conservation laws such as Total Lagrangian, Eulerian and Updated Reference Lagrangian. Hyperbolicity of the system of conservation laws will be shown and the accurate wave speed bounds will be presented, the latter critical to ensure stability of explicit time integrators. For spatial discretisation, a vertex‐based Finite Volume method is employed and suitably adapted. To guarantee stability from both the continuum and the semi‐discretisation standpoints, an appropriate numerical interface flux (by means of the Rankine–Hugoniot jump conditions) is carefully designed and presented. Stability is demonstrated via the use of the time variation of the Hamiltonian of the system, seeking to ensure the positive production of numerical entropy. A range of three dimensional benchmark problems will be presented in order to demonstrate the robustness and reliability of the framework. Examples will be restricted to the case of isothermal reversible elasticity to demonstrate the potential of the new formulation.

A gas-mechanical-damage coupling model based on the TLF-SPH method and its application to gas seepage in fractured coal

The coupled calculation of gas seepage in fractured coal is a challenge in computational mechanics. In this paper, a gas seepage model of fractured coal is developed based on smoothed particle hydrodynamics (SPH). Then, the accuracy of the seepage model based on the partial differential equation (PDE) with a constant coefficient is verified by a reference example. Moreover, the seepage characteristics of gas in homogeneous coal based on a PDE with constant and variable coefficients are analyzed, and their numerical errors are compared. Then, based on the gas seepage model of fractured coal, the effects of heterogeneity, defect opening and inclination on gas seepage in coal are studied. In addition, considering the influence of coal deformation and preexisting defects on gas seepage, a gas-mechanical-damage coupling model for gas seepage simulation is proposed by using the SPH method with the total Lagrangian formula (TLF-SPH). Subsequently, the permeability evolution and gas seepage characteristics of borehole surrounding rock under different conditions (such as overburden load, extraction pressure, borehole radius and initial gas pressure) are investigated, revealing the seepage-mechanical coupling mechanism of fractured coal during gas extraction. This work provides important theoretical support for mine gas extraction and gas outburst disaster prevention technology.

Isogeometric analysis of hyperelastic Timoshenko beams under large deformation and incompressibility constraints

This article investigates the nonlinear analysis of hyperelastic planar curved beams based on the total Lagrangian Timoshenko theory. The fully incompressible neo-Hookean model has been selected as the constitutive relation to accurately capture large deformations. The nonlinear equations and boundary conditions derived from the virtual work principle are solved using the Newton–Raphson method. Employing isogeometric analysis with NURBS functions to discretize the unknown kinematics, including displacements and rotation, and the linearized governing equations. The study accounts for conservative and non-conservative forces, validating results against available experimental and numerical results reported in the literature, confirming the proposed formulation’s effectiveness and precision.

Harmonic in-plane patch loading induced nonlinear dynamic instability of unidirectional and bidirectional stiffened composite plates

A finite element framework is established to study the nonlinear dynamic instability of stiffened plates subjected to in-plane harmonic patch loading. An eight-noded isoparametric degenerated shell element and a three-noded curved beam element are used to model the skin and the stiffener, respectively. The Green–Lagrange strain displacement relationship is adopted to formulate the system matrices for the plate and the stiffener, adopting the total Lagrangian approach. The Bolotin method is adopted to trace the boundaries of linear dynamic instability region. The Incremental Harmonic Balance (IHB) method is used to investigate the nonlinear instability behavior of stiffened plates. The effect of varying loading patches along with the number of stiffeners in unidirectional ( x-directional and y-directional) and bidirectional ( x as well as y-directional) stiffened plates on the nonlinear frequency response is studied. It is observed that the cross-stiffened plates with the higher number of stiffeners are dynamically more stable than the unidirectional stiffened plates. Moreover, the stiffened plates with smaller loading patches are dynamically more stable than those with larger loading patches.

Sequential formulation of all‐way coupled finite strain thermoporomechanics for largely deformable gas hydrate deposits

AbstractWe develop a numerically stable sequential formulation of thermoporomechanics for largely deformable gas hydrate deposits, extended from the fixed stress split of infinitesimal transformation. Constitutive equations are based on the total Lagrangian approach for both flow and geomechanics, including dynamic full tensor permeability and thermal conductivity updated from the deformation gradient. For space discretization, we take the cell‐centered finite volume and node‐based finite element method for flow and geomechanics, respectively. Then, we propose a sequential implicit method for all‐way coupled thermoporomechanics, where the nonisothermal multiphase flow problem of gas hydrates is solved implicitly first and then the geomechanics problem is solved implicitly at the next step. During solution of the flow problem, we fix the rate of first Pioal total stress for numerical stability as well as apply porosity correction and entropy correction to account for geomechanical effects. We test numerical examples where flow and geomechanics parameters are based on deep oceanic gas hydrate deposits. When applying depressurization, even though the results between the infinitesimal transformation and finite strain geomechanics are similar in the early stages due to small deformation, we find differences between them in the late times as deformation becomes large. Accordingly, permeability and thermal conductivity tensors become nonisotropic full tensors although they are initially isotropic. We identify numerical stability of the developed sequential method from the test cases that exhibit the highly complex coupled gas hydrate systems with large deformation. Thus, the proposed sequential formulation can be applied in largely deformable gas hydrate systems.