Total Lagrangian Formulation Research Articles

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.

A modular finite element approach to saturated poroelasticity dynamics: Fluid–solid coupling with Neo-Hookean material and incompressible flow

Several methods have been developed to model the dynamic behavior of saturated porous media. However, most of them are suitable only for small strain and small displacement problems and are built in a monolithic way, so that individual improvements in the solution of the solid or fluid phases can be difficult. This study shows a macroscopic approach through a partitioned fluid–solid coupling, in which the skeleton solid is considered to behave as a Neo-Hookean material and the interstitial flow is incompressible following the Stokes–Brinkman model. The porous solid is numerically modeled with a total Lagrangian position-based finite element formulation, while an Arbitrary Lagrangian-Eulerian stabilized finite element approach is employed for the porous medium flow dynamics. In both fields, an averaging procedure is applied to homogenize the problem, resulting in a macroscopic continuous phase. The solid and fluid homogenized domains are overlapped and strongly coupled, based on a block-iterative solution scheme. Two-dimensional simulations of wave propagation in saturated porous media are employed to validate the proposed formulation through a comprehensive comparison with analytical and numerical results from the literature. The analyses underscore the proposed formulation as a robust and precise modular approach for addressing dynamic problems in poroelasticity.

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.

Accelerating aeroelastic UVLM simulations by inexact Newton algorithms

We 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.

Coupled peridynamic modeling of hydraulic fracturing in rocks

This paper presents a fully coupled peridynamics method for simulating hydraulic fracturing in rocks. The proposed method involves a rigorous coupling between the classical total-Lagrangian formulation and a new semi-Lagrangian formulation within the framework of peridynamics. The total-Lagrangian formulation is employed to simulate a rock subjected to fluid-driven fracturing, whereas the semi-Lagrangian formulation is used to solve the Navier–Stokes equations for fluids in a nonlocal manner. A coupling algorithm is developed between the two formulations to model fluid–solid interactions without causing unphysical penetration. The proposed method was validated through a simulation of a water column collapse problem. It was further validated using a Kristianovich–Geertsma–De Klerk (KGD) fracturing problem, where a plane strain fracture propagates owing to fluid injection. The proposed method reasonably captures the fracturing pattern, fracture propagation speed, and injection pressure better than the analytical solution. It offers a unified framework within the peridynamics theory for modeling solid, fluid, and fluid-driven fractures.

Contact framework for total Lagrangian smoothed particle hydrodynamics using an adaptive hybrid kernel scheme

AbstractTotal Lagrangian SPH (TLSPH) offers many advantages that can overcome instability and computational time issues. However, it is not naturally applicable to problems with contact. Many problems in solid dynamics and structural analysis require contact definition. In order to use the total Lagrangian formulation, further development for the contact algorithm is required. This research proposes an algorithm to implement inter‐particle contact in TLSPH using an adaptive hybrid‐kernel scheme for the first time. A combination of the updated and total Lagrangian formulations at the contact surface is introduced, in which the kernel type change automatically when particles from a different body are detected. In kernel change from the updated to the total Lagrangian formulation, the gradient correction cannot be performed using a common correction method because different frames of reference are used. Therefore, a hybrid‐kernel correction technique is proposed to maintain the zeroth‐ and first‐order completeness. Then, the contact force is obtained from the inter‐particle force using the conservation of momentum. The proposed method does not require a user‐defined parameter that makes the application straightforward. Based on the test, the proposed method agrees well with the data available in published literature and the finite element method. Compared with the contact method in classical SPH, the proposed TLSPH shows a significant stability improvement.

Ordinary state-based peridynamic model for out-of-plane deformation and damage analysis of composite laminates

An ordinary state-based peridynamic (PD) model for predicting out-of-plane mechanical behaviour of composite laminates has been proposed, in which three types of PD bonds are used to describe the reinforcement properties. The non-local strain energy density and peridynamic formulations are obtained based on the principle of virtual work by using Total Lagrange formulation, and the force density is reformulated by interpolation technique. Since the Mindlin plate is regarded as the research object of this PD model, the transverse shear deformation of laminates is considered. Also, the dilatation term is included in the force density vector, and flaws in Poisson's ratio of materials can be overcome. In the proposed PD, the critical strain energy density rather than the critical curvature is adopted as the failure criterion. The capability of the developed PD model was demonstrated by the bending examples of composite laminates with different fiber orientations, and damage analysis was further conducted to demonstrate the strong capability of proposed PD model in replicating the damage process of laminates. In addition, a single-layer material point model (SLMPM) can be implemented in the proposed PD algorithm, and the computational efficiency of numerical models will be greatly improved.

Vibration of solid and thin-walled slender structures made of soft materials by high-order beam finite elements

In this work, high-order beam (1D) finite element models for the modal analysis of structures made of compressible and nearly-incompressible hyperelastic soft materials are presented in the well-established framework of Carrera Unified Formulation (CUF). In this investigation, the modal behavior of soft structures subjected to progressively increasing loads can be correctly predicted by higher-order structural theories, and the influence of pre-stress conditions applied on the modal response of structures is investigated. The mathematical formulation of hyperelastic isotropic materials is presented in terms of invariants of the right Cauchy–Green strain tensor, obtaining the most general expression of the Piola–Kirchoff 2 stress tensor and tangent elasticity tensor, both independent of the model adopted for the material strain energy function. Governing equations in matrix forms for the static nonlinear analysis and subsequent vibration problem around non-trivial equilibrium states are derived through the Principle of Virtual Displacements (PVD) under a total Lagrangian formulation, defining the fundamental nuclei of stiffness matrices and internal and external forces vector, all independent of expansion theories and kinematic models adopted in the mathematical modeling of finite elements. Actual numerical results are obtained by an iterative Newton–Raphson linearized scheme coupled with line-search algorithms, and they are compared with results obtained by the commercial code ABAQUS. Our proposed models are tested with large strain problems involving hyperelastic slender and thin-walled structures, for which mode aberration such as crossing or bearing are observed.

Large Deformation Analysis of Hyperelastic Continuum with Hexahedral Adaptive Finite Elements

The use of hyperelastic materials capable of large deformations, such as elastomeric bearings used to reduce seismic effects, is quite common in civil engineering. Such environments are, in most cases, addressed by numerical solution techniques such as the finite element method. In case of large deformations, nonlinear analysis is used in the solution. In the study presented here, large deformations of a hyperelastic continuum expressed by the Mooney-Rivlin material model are calculated using hexahedral adaptive finite elements. A code was written in MATLAB using the total Lagrangian formulation for the nonlinear adaptive finite element solution. Comparisons were made with Abaqus software to check the consistency of the results obtained from this program. It has been observed that local refinements in the adaptive element mesh occur in the regions where they are needed. Considering the variation of maximum displacement and maximum stress with the number of elements, it has been observed that mesh refinement creates a convergent solution.

Coupled total- and semi-Lagrangian peridynamics for modelling fluid-driven fracturing in solids

This paper presents a novel computational approach for modelling fluid-driven fracturing in quasi-brittle solids using peridynamics. The approach leverages a rigorous coupling of the total- and semi-Lagrangian formulations of peridynamics. Specifically, the total-Lagrangian formulation, whch is based on classical peridynamics theory used for modelling fractures in solids, is combined with the semi-Lagrangian formulation to solve the Navier-Stokes equation using a non-local differential operator for weakly compressible fluid flow. The proposed approach offers a unified peridynamics-based framework that enables simulations of a wide range of fluid-driven fracturing problems in solids. The framework can model the solid as either an ordinary or non-ordinary material and quantify the fluid with different equations of state. To prevent unphysical inter-penetration, a fluid-solid interaction scheme that assumes fictitious fluid points in the solid domain is proposed to quantify the forces at the fluid-solid interface. The proposed computational approach is validated through simulations of several classic problems, including the dam break and the Kristianovich-Geertsma-de Klerk (KGD) problem. The predictive capability of the proposed approach is further demonstrated by numerical examples of fractures induced by fluid injection in sandstone, which reasonably capture the fracture patterns compared to experimental results. The presented approach offers an alternative to explicit modelling of fluid-driven fracturing processes and may find wide and useful applications to a variety of industrial and geophysical processes.

Analysis of transversely isotropic compressible and nearly-incompressible soft material structures by high order unified finite elements

This work proposes high-order beam (1D) and plate (2D) finite element models for the large strain analysis of compressible and incompressible transversely isotropic hyperelastic media, defined within the Carrera Unified Formulation (CUF) framework. The strain energy density function adopted in fiber-reinforced hyperelastic materials modeling is presented and expressed in terms of invariants and pseudo-invariants of the right Cauchy-Green strain tensor. The explicit expression of the tangent elasticity tensor is derived through the assumption of coupled formulation of strain energy functions. Refined fully nonlinear beam and plate models are defined in a total Lagrangian formulation, deriving the governing equations of the nonlinear static analysis through the Principle of Virtual Displacements in terms of fundamental nuclei, in resulting expressions of internal and external forces vectors, and tangent stiffness matrix independent of kinematic models and approximation theories adopted. The iterative Newton-Raphson linearization scheme coupled with the arc-length constraint is adopted to obtain actual numerical solutions. Different benchmark analyses in hyperelasticity are performed to assess the capabilities of our proposed model, analyzing the three-dimensional stress field for moderate to large strain states and comparing actual numerical results with exact closed-form solutions or results available in the literature, demonstrating the capabilities and reliability of CUF models in the analysis of fiber-reinforced soft materials and structures.

High fidelity nonlinear finite element tire modeling for dynamic analysis: Total Lagrangian formulation in rolling contact

Tire design with respect to vibrational behavior and noise emission is a challenging task due to the large number of parameters and their coupling. Therefore, accurate high fidelity nonlinear Finite Element (FE) models are needed to aid in the design process. The widely used Arbitrary Lagrangian Eulerian (ALE) models, despite their computational efficiency, are limited to rib-slicked tires and require significant effort to consider inelastic material behavior and frictional phenomena. To overcome these limitations a novel nonlinear FE tire model is presented based on the Total Lagrangian (TL) formulation. The TL model is compared to the reference ALE model for the case of stationary rolling on a smooth and a rough road surface. The observed match of the TL model with respect to the hub and contact forces proves the feasibility of using the TL formulation in the context of rolling contact tire simulations. Therefore, the TL model can be seen as a valid alternative to the frequently used ALE models.

ADER discontinuous Galerkin Material Point Method

AbstractThe first‐order accurate discontinuous Galerkin Material Point Method (DGMPM), initially introduced by Renaud et al. (J Comput Phys. 2018;369:80–102.), considers a solid body discretized by a collection of material points carrying the history of the matter, embedded in an arbitrary grid on which a nodal discontinuous Galerkin approximation is defined, and that serves to solve balance equations. This method has been shown to be promising, especially for solving hyperbolic problems in finite deforming solids (Renaud et al. Int J Numer Methods Eng. 2020;121(4):664–689. Renaud et al. Comput Methods Appl Mech Eng. 2020;365:112987.). The main goal of this research is to extend the first‐order DGMPM to arbitrary high‐order accurate approximations. This is performed by adapting the ADER (Arbitrary high order DErivative Riemann problem) approach (Busto et al. Front Phys. 2020;8:32.) to the particular spatial discretization of the DGMPM. First, the predictor step permits to design a particle‐to‐grid projection of arbitrary high order of accuracy, consistent with that of the nodal discontinuous Galerkin approximation defined on the arbitrary grid. This is performed using a moving least square approximation for the ADER predictor field. Second, since the degrees of freedom of the predictor field are now defined at material points, the computation of the constitutive response of the material is ensured to be always performed at these material points. This is of crucial importance for history‐dependent constitutive models because it avoids any diffusive transfer of internal variables on a new computational grid. Finally, a total Lagrangian formulation of equations is kept, which allows to precompute once and for all both the nodal discontinuous Galerkin approximation and that of the ADER predictor field, until the arbitrary grid is discarded if required. The method is illustrated on a few two‐dimensional numerical examples, on which comparisons are shown with the ADER‐DGFEM and Runge–Kutta‐DGFEM.

Nonunit quaternion parametrization of a Petrov–Galerkin Cosserat rod finite element

AbstractThe application of the Petrov–Galerkin projection method in Cosserat rod finite element formulations offers significant advantages in simplifying the expressions within the discrete virtual work functionals. Moreover, it enables a straight‐forward and systematic exchange of the ansatz functions, specifically for centerline positions and cross‐section orientations. In this concise communication, we present a total Lagrangian finite element formulation for Cosserat rods that attempts to minimize the number of required theoretical concepts. The chosen discretization preserves objectivity and allows for large displacements/rotations and for large strains. The orientation parametrization with nonunit quaternions results in a singularity‐free formulation.

Effect of different material on the folding of composite ultrathin tape spring hinges via unifed 2D shell models

The aim of this work is to analyze the effect of different materials on the folding of ultrathin tape spring hinges. This mechanism is modeled using the Finite Element Method (FEM) combined with the Carrera Unified Formulation (CUF) for the development of refined 2D shell models. These models can deal with different materials and different theoretical approximations can be introduced in an automatic way thanks to the hierarchical capability of the CUF formalism, according to which the degree of refinement of the mathematical model is an input to the analysis. The nonlinear governing equations are developed using a Total Lagrangian formulation and solved via a Newton-Raphson linearization technique with arc-length type constraints. A validation of the proposed model is carried out by comparing the results with those available in the literature involving a cured T300-1k/913 tow and HexPly 913 resin. After a convergence analysis, the mathematical model proves to ensure a good fit. This model is used for the final analysis with different materials, i.e. isotropic metallic with hardened steel tape spring and unidirectional T300 graphite fibre/epoxy prepreg with 34% resin content by weight. The results, in the form of moment angle curves, are reported for two different values of the thickness of the structure.

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.

Evaluations of atomic-resolution strain fields using molecular dynamics simulations combined with corrected smoothed particle hydrodynamics

Deformation analyses of polymeric materials using molecular dynamics (MD) simulations are very useful in evaluating the physical properties based on the atomic features. Generally, the average responses of a system, such as the stress–strain response, are evaluated, but a molecular chain around a solid should deform in an unusual manner in a composite system, such as a solid/polymer system. This may lead to damage and a microscopic inelastic deformation, ultimately affecting the macroscopic behavior of the material. In this study, we propose a technique for use in evaluating the local pseudo-continuum deformation quantities, such as strain and stretch, during MD simulations by introducing deformation analysis via corrected smoothed particle hydrodynamics in a total Lagrangian formulation. Three different types of systems (uniform deformation of an Au crystal, cured resin sandwiched between silica walls, and cured resin containing a nanofiller) are considered, and the unique local deformations in the composite systems are quantitatively evaluated. In particular, local constitutive behavior can be evaluated using this technique, and in the sandwiched system, a high correlation is observed between the local stiffness inhomogeneity and the cohesive failure point. In the filler-containing system, a highly rigid epoxy resin layer is observed around the filler, and local deformation at the poles of the filler is confirmed owing to elongation. The observed trends are consistent with the experimental results. The insights obtained should contribute to the evaluation and understanding of microscopic deformation and damage, which is the beginning of macroscopic failure in a composite material.

Preface to the topical issue on applied and nonlinear dynamics: Part II

The current special issue of the GAMM Mitteilungen, which is the second of a two-part series, contains several contributions on the topic of Applied and Nonlinear Dynamics. We are very happy that several teams of authors have accepted our invitation to report on recent developments, research highlights and emerging application areas in Applied and Nonlinear Dynamics. This second part of the topical issue on Applied and Nonlinear Dynamics includes five interesting papers. These are devoted to numerical and experimental methods in applied and nonlinear dynamics as well as advanced applications of multibody systems and optimal control methods to dynamical systems. Contribution 3 deals with stationary solutions in applied dynamics. Thereby a unified framework for the numerical calculation and stability assessment of periodic and quasi-periodic solutions based on invariant manifolds is presented. Paper 2 gives an overview of dynamic human body models in vehicle safety, a unique application of multibody dynamics. In paper 1, a family of total Lagrangian Petrov-Galerkin Cosserat rod finite element formulations is presented. Paper 5 discusses continuation methods for lab experiments of nonlinear vibrations. Finally paper 4 deals with the optimal operation of dielectric elastomer wave energy converters under harmonic and stochastic excitation.

A discontinuous Galerkin formulation for nonlinear analysis of multilayered shells refined theories

A novel pure penalty discontinuous Galerkin method is proposed for the geometrically nonlinear analysis of multilayered composite plates and shells, modelled via high-order refined theories. The approach allows to build different two-dimensional equivalent single layer structural models, which are obtained by expressing the covariant components of the displacement field through-the-thickness via Taylor’s polynomial expansion of different order. The problem governing equations are deduced starting from the geometrically nonlinear principle of virtual displacements in a total Lagrangian formulation. They are addressed with a pure penalty discontinuous Galerkin method using Legendre polynomials trial functions. The resulting nonlinear algebraic system is solved by a Newton–Raphson arc-length linearization scheme. Numerical tests involving plates and shells are proposed to validate the method, by comparison with literature benchmark problems and finite element solutions, and to assess its features. The obtained results demonstrate the accuracy of the method as well as the effectiveness of high-order elements.

Accelerated nonlinear finite element method for analysis of isotropic hyperelastic materials nonlinear deformations

In this work, an approximate Jacobian matrix is proposed based on the total Lagrangian formulation of Finite Element Method for isotropic hyperelastic materials. The approximate Jacobian matrix can take the place of the exact Jacobian matrix in the Newton–Raphson method to avoid frequent construction and factorization of the Jacobian matrix. A new Quasi–Newton method employing the approximate Jacobian matrix is developed to significantly improve the convergence rate. The proposed method was tested on three examples with the combinations of three hyperelastic material models and three element types. The results show that the proposed method is more efficient than ABAQUS and CALCULIX (an open-source FEM software package) on all of the tests without loss of accuracy. It is up to 100 times faster than the traditional Quasi–Newton method, and at least 2.5 times faster than ABAQUS.