Updated Lagrangian Formulation Research Articles

Updated Lagrangian finite element formulations of various biological soft tissue non-linear material models: a comprehensive procedure and review

Simplified material models are commonly used in computational simulation of biological soft tissue as an approximation of the complicated material response and to minimize computational resources. However, the simulation of complex loadings, such as long-duration tissue swelling, necessitates complex models that are not easy to formulate. This paper strives to offer the updated Lagrangian formulation comprehensive procedure of various non-linear material models for the application of finite element analysis of biological soft tissues including a definition of the Cauchy stress and the spatial tangential stiffness. The relationships between water content, osmotic pressure, ionic concentration and the pore pressure stress of the tissue are discussed with the merits of these models and their applications.

Nonlinear response of elastic cables with flexural-torsional stiffness

A geometrically exact mechanical formulation is proposed to describe three-dimensional motions of flexible cables without any restriction on the amplitude of such motions. The nonlinear equations of motion are formulated via an updated Lagrangian formulation taking the prestressed catenary equilibrium under gravity as the start configuration. By employing a single space coordinate parametrization, the kinematics feature finite displacement and rotation of the cable cross sections, assumed rigid in their own planes. The ensuing generalized strain parameters and curvatures retain the full geometric nonlinearities. By considering several case-study cables within the groups of taut and shallow cables, nonlinear equilibrium analyses are performed to investigate the effects of the cable bending and torsional stiffness for nontrivial boundary conditions. The full nonlinear formulation allows to estimate the size of the boundary layers as well as the stress states within them otherwise unknown adopting classical theories of purely stretchable cables.

Discrete-Mindlin finite element for nonlinear geometrical analysis of shell structures

A finite degenerated shell element based on Mindlin discrete approach is presented for nonlinear geometric analysis in large displacements and small deformations. The element, called DMQS (Discrete Mindlin Quadrilateral Shell) with four-nodes and 6 dof/node, includes a constant transverse displacement and rich quadratic rotations. Since the global kinematics of shell is dominated by a rigid body rotation and motion involving small displacements and rotations, an Updated Lagrangian Formulation at each Iteration is used. The tangent stiffness matrix has been treated as a sum of two parts: linear part related to the degenerated shell model (DMQS) as well as the initial stress part associated with the well-known membrane standard element. Some standard nonlinear benchmarks are presented herein, where very good results have been obtained and show better accuracy than similar quadrilateral elements.

Effects of Process Parameters on the Formability of Miniature Layered Cups

Based on materials, different punch radii (0.3, 0.35, 0.4, 0.45, and 0.5 mm), two sets of diameter-diameter ratio 1.(.167, 1.25, 1.33, 1.4167, and 1.5) and 2.(1.6, 1.45, 1.33, 1.231, and 1.143), and two sets of depth ratio 1.(1.3, 1.4, 1.5, 1.6, and 1.7) and 2.(2.14, 1.875, 1.67, 1.5, and 1.36) are used for the stamping processes to analyze the simulation and experimental difference in copper sheet-metal (C1100) miniature layered cups. Prandtl-Reuss flow rule is integrated with finite deformation theory and Updated Lagrangian Formulation (ULF) to establish the incremental elastic-plastic deformation Finite Element Method in Coulomb’s Friction Law for simulating the miniature layered cup process. Generalized rmin algorithm is utilized in the forming process for dealing with elastic-plastic behaviors and die contact. From the simulation data, the relationship among deformation history, punch load, and punch stroke, the stress-strain distribution, and the distribution of the thinnest thickness by different punch radii are acquired.

A beam formulation for large displacement analysis of composite frames with semi-rigid connections

This paper presents a geometrically non-linear beam formulation for large displacement analysis of composite beam type structures with semi-rigid connections. Within the framework of updated Lagrangian incremental formulation and the nonlinear displacement field of thin-walled cross-sections, which accounts for restrained warping and the second-order displacement terms due to large rotations, the equilibrium equations of a straight beam element are firstly developed. Due to the nonlinear displacement field, the geometric potential of semitangential moment is obtained for both the internal torsion and bending moments, respectively. To account for the semi-rigid connection behaviour, a special transformation procedure is developed. The laminates are modelled on the basis of classical lamination theory. In order to illustrate the application of the proposed formulation, several numerical examples are presented.

Effect of micro-textured tools on machining of Ti–6Al–4V alloy: An experimental and numerical approach

In this work, an attempt is made to reduce the detrimental effects that occurred during machining of Ti–6Al–4V by employing surface textures on the rake faces of the cutting tools. Numerical simulation of machining of Ti–6Al–4V alloy with surface textured tools was employed, taking the work piece as elasto-plastic material and the tool as rigid body. Deform 3D software with updated Lagrangian formulation was used for numerical simulation of machining process. Coupled thermo-mechanical analysis was carried out using Johnson-cook material model to predict the temperature distribution, machining forces, tool wear and chip morphology during machining. Turning experiments on Ti–6Al–4V alloy were carried out using surface textured tungsten carbide tools with micro-scaled grooves in preferred orientation such as, parallel, perpendicular and cross pattern to that of chip flow. A mixture of molybdenum disulfide with SAE 40 oil (80:20) was used as semi-solid lubricant during machining process. Temperature distribution at tool–chip interface was measured using an infrared thermal imager camera. Feed, thrust and cutting forces were measured by a three component-dynamometer. Tool wear and chip morphology were captured and analyzed using optical microscopic images. Experimental results such as cutting temperature, machining forces and chip morphology were used for validating numerical simulation results. Cutting tools with surface textures produced in a direction perpendicular to that of chip flow exhibit a larger reduction in cutting force, temperature generation and reduced tool wear.

Postbuckling behavior of flexibly connected single layer steel domes

In the construction of single layer domes and vaulted structures manufactured ball-node type connections are widely used. These ball-nodes deform due to the action of a three-dimensional state of stress while offering their resistance. These deformations result in a loss of connection stiffness, which influences the overall stability of these domes, and is usually ignored in the analysis. This work presents an investigation on the effect of ball-node deformations in the inelastic post-buckling behavior of domes and vaulted structures. A model for the load-deformation properties of the ball-nodes has been proposed in this study, in the lines of EC3-8. This connection flexibility model is integrated into a corotated–updated Lagrangian finite element formulation. Results for single layer domes and a braced barrel vault are presented. Although the present investigation deals with ball nodes only, the computational framework is still relevant for other type of connections of space structures, with modifications.

Finite strain discrete dislocation plasticity in a total Lagrangian setting

We present two total Lagrangian formulations for finite strain discrete dislocation plasticity wherein the discrete dislocations are presumed to be adequately represented by singular linear elastic fields thereby extending the superposition method of Van der Giessen and Needleman (1995) to finite strains. The finite deformation effects accounted for are (i) finite lattice rotations and (ii) shape changes due to slip. The two formulations presented differ in the fact that in the “smeared-slip” formulation the discontinuous displacement field is smeared using finite element shape functions while in the “discrete-slip” formulation the weak form of the equilibrium statement is written to account for the slip displacement discontinuity. Both these total Lagrangian formulations use a hyper-elastic constitutive model for lattice elasticity. This overcomes the issues of using singular dislocation fields in a hypo-elastic constitutive relation as encountered in the updated Lagrangian formulation of Deshpande et al. (2003). Predictions of these formulations are presented for the relatively simple problems of tension and compression of single crystals oriented for single slip. These results show that unlike in small-strain discrete dislocation plasticity, finite strain effects result in a size dependent tension/compression asymmetry. Moreover, both formulations give nearly identical predictions and thus we expect that the “smeared-slip” formulation is likely to be preferred due to its relative computational efficiency and simplicity.

Investigation of wave propagation in double cylindrical rods considering the effect of prestress

This paper presents the investigation of guided wave propagation in prestressed double-cylinder structure. Based on Hertzian contact theory, the interaction between two rods is treated as a plane strain problem and the stress state in the waveguide under the static load is obtained. The stress state is considered as a prestressed configuration for elastic wave propagation analysis in double cylindrical rods. The elastodynamic equation of the prestressed structure is established with the updated Lagrangian formulation and the wave finite element (WFE) method. Firstly, the equation is verified by the application on an aluminum rod compared with the Euler–Bernoulli beam theory. Then, dispersion curves for single rod and double cylindrical rods without prestress are computed. Besides, at the dimensionless frequency 0.25 the propagating modes in double cylindrical rods are identified with mode shapes and displacement vectors. The guided waves in double rods consist of the modes different from single rod. Particularly, there exist two kinds of torsional-like modes, one of which has a twist center and the other has two. The latter changes obviously with the increase of prestress in the waveguide. At low frequencies, torsional-like modes are very sensitive to the variation of prestress; in addition, the prestress configuration has little influence on propagating modes at mid-frequencies but some at high frequencies.

Nonlinear Equilibrium and Stability Analysis of Axially Loaded Piles Under Bilateral Contact Constraints

This paper presents a nonlinear stability analysis of piles under bilateral contact constraints imposed by a geological medium (soil or rock). To solve this contact problem, the paper proposes a general numerical methodology, based on the finite element method (FEM). In this context, a geometrically nonlinear beam-column element is used to model the pile while the geological medium can be idealized as discrete (spring) or continuum (Winkler and Pasternak) foundation elements. Foundation elements are supposed to react under tension and compression, so during the deformation process the structural elements are subjected to bilateral contact constraints. The errors along the equilibrium paths are minimized and the convoluted nonlinear equilibrium paths are made traceable through the use of an updated Lagrangian formulation and a Newton-Raphson scheme working with the generalized displacement technique. The study offers stability analyses of three problems involving piles under bilateral contact constraints. The analyses show that in the evaluation of critical loads a great influence is wielded by the instability modes. Also, the structural system stiffness can be highly influenced by the representative model of the soil.

Uncertain-but-Nonrandom Dynamic Optimization of Composites Flexible Membrane Structure

In the past few years, owing to the vital role in maritime safety monitoring and marine hazard warning, stratospheric airship has become a research hotspot around the world. The structure of stratospheric airship is a typical aerated flexible membrane one. It is geometrical nonlinear in essence and it bears a distinct mechanical nature of large deformation. Therefore it makes optimizing the structure more difficult. The current studies of air-supported flexible membrane structural optimization ignore the uncertainty influence which is resulted from the loading environments, material properties and structural parameters. According to the Lagrangian Coordinate method, the deformed configuration is used as a reference in order to build accurate strain/displacement geometry. The Equilibrium equation is created by using the principle of virtual work. Meanwhile, the accuracy and stability of the solution is ensured by using the Updated Lagrangian Formulation. The uncertain parameters which are introduced during the analysis and optimization of the aerated flexible membrane structure are portrayed by interval mathematical theory. With respect to engineering application, the factors of uncertainty are considered. Maximizing the structural fundamental frequency (or first-order natural frequency) is taken as the objective, while structural integrity is taken as the constraint. Then the dynamic optimization is carried out under interval uncertainty. In this way the reliability of the membrane structure in complex conditions is improved and the probability of failure is reduced.

Elasto-Plastic Finite Deformation Simulations using 3-D Parallel XFEM

In this work, a parallel XFEM is presented to solvenonlinear solid mechanics problems. The material behavior is modeled using isotropic hardening based on J2 theory of plasticity. The geometrical nonlinearity is modeled through updated Lagrangian formulation. Penalty approach is employed to impose the contact constraint at the material interfaces. A node-to-node technique is used to model the contact behavior. Aparallel algorithm is implemented to reduce the computational time. Some nonlinear problems are simulated to check the efficiency of the parallel algorithm.

NONLINEAR ANALYSIS OF VISCOELASTICALLY LAYERED ROLLS IN STEADY STATE ROLLING CONTACT

The present paper analyzes the steady state rolling contact (SSRC) response of nonlinear viscoelastically layered rigid roll indented by a rigid cylindrical indenter. Both material and geometrical nonlinearities are accounted for in the framework of the updated Lagrangian finite element formulation. The Schapery's viscoelastic creep model is adopted to model the viscoelastic behavior. To accommodate the steady state rolling condition, the constitutive equations are recast into a spatially dependent incremental form. Throughout the contact interface, the Lagrange multiplier method is used to enforce the contact constraints, while the classical Coulomb's law is adopted to simulate friction. The resulting nonlinear equilibrium equations are solved by the Newton–Raphson method. The developed model is applied to analyze a viscoelastically layered rigid roll in steady state rolling and intended by a rigid cylindrical indenter. Results showed the distinct effects of angular velocity, retardation time, indenter radius, and viscoelastic layer thickness on the SSRC configuration.

Assumed Natural Strain NURBS-based solid-shell element for the analysis of large deformation elasto-plastic thin-shell structures

In this work, a recently proposed quadratic NURBS-based solid-shell element based on the Assumed Natural Strain (ANS) method is applied in the analysis of shell-like structures in the geometrical nonlinear regime, together with small strain plasticity. The proposed formulation is based on the additive split of the Green–Lagrange strain tensor, leading to a straightforward implementation of the nonlinear kinematics and to the introduction of a corotational coordinate system, used to integrate the constitutive law, ensuring incremental objectivity. Since the proposed approach is based on Updated Lagrangian formulation combined with a corotational coordinate system, the extension of the ANS methodology is straightforward. Well-known benchmark tests are employed to assess the performance of the proposed formulation and to establish a detailed comparison with the formulations available in the literature. The results indicate that the proposed solid-shell approach based on the NURBS ANS methodology presents good predictability characteristics in the analysis of elasto-plastic thin-shell structures subjected to large deformations.

Buckling analysis of thin-walled functionally graded sandwich box beams

Buckling analysis of thin-walled functionally graded (FG) sandwich box beams is investigated. Material properties of the beam are assumed to be graded through the wall thickness. The Euler-Bernoully beam theory for bending and the Vlasov theory for torsion are applied. The non-linear stability analysis is performed in framework of updated Lagrangian formulation. In order to insure the geometric potential of semitangental type for internal bending and torsion moments, the non-linear displacement field of thin-walled cross-section is adopted. Numerical results are obtained for FG sandwich box beams with simply–supported, clamped–free and clamped–clamped boundary conditions to investigate effects of the power-law index and skin-core-skin thickness ratios on the critical buckling loads and post-buckling responses. Numerical results show that the above-mentioned effects play very important role on the buckling analysis of sandwich box beams.

Updated lagrangian mixed finite element formulation for quasi and fully incompressible fluids

We present a mixed velocity---pressure finite element formulation for solving the updated Lagrangian equations for quasi and fully incompressible fluids. Details of the governing equations for the conservation of momentum and mass are given in both differential and variational form. The finite element interpolation uses an equal order approximation for the velocity and pressure unknowns. The procedure for obtaining stabilized FEM solutions is outlined. The solution in time of the discretized governing conservation equations using an incremental iterative segregated scheme is described. The linearization of these equations and the derivation of the corresponding tangent stiffness matrices is detailed. Other iterative schemes for the direct computation of the nodal velocities and pressures at the updated configuration are presented. The advantages and disadvantages of choosing the current or the updated configuration as the reference configuration in the Lagrangian formulation are discussed.

A coupled Eulerian–Lagrangian extended finite element formulation for simulating large deformations in hyperelastic media with moving free boundaries

We present a coupled Eulerian–Lagrangian (CEL) formulation aimed at modeling the moving interface of hyperelastic materials undergoing large to extreme deformations. This formulation is based on an Eulerian description of kinematics of deformable bodies together with an updated Lagrangian formulation for the transport of the deformation gradient tensor. The extended finite element method (XFEM) is used to discretize the mechanical equilibrium and deformation gradient transport equations in a two-phase domain. A mixed interpolation scheme (biquadratic for the velocity and bilinear for the deformation gradient) is adopted to improve the accuracy of the numerical formulation. The interface describing the deformed shape of the body is represented by the level set function and is evolved using the grid based particle method. The performance of the scheme is explored in two-dimensions in the compressible regime. For an adequate spatial and temporal discretization, our numerical results are in good agreement with theory and with numerical results from the traditional Lagrangian formulation (in Abaqus). The advantage of the proposed formulation is that material motion is not coupled with that of the mesh; this eliminates the issues of mesh distortion and the need for remeshing associated with Lagrangian formulations when bodies undergo very large distortions. It is therefore well adapted to describe the motion of complex fluids and soft matter whose physical properties are intermediate between conventional liquids and solids.

NUMERICAL ANALYSIS OF ELECTROMAGNETO-MECHANICAL COUPLING USING LAGRANGIAN APPROACH AND ADAPTIVE TIME STEPPING METHOD

Electromagneto-mechanical coupling effect associated with vibration induced eddy currents plays an important role in the evaluation and design of many magnetic devices, such as a tokamak. In this paper, a updated Lagrangian formulation for eddy current analysis of deformable bodies was presented through transformation from reference configuration to current configuration. A coupled analysis method based on Lagrangian meshes and Ar method was developed. A corresponding code was developed and verified by simulating Team Problem 16. The numerical method was further updated by introducing an adaptive time stepping procedure to improve its simulation efficiency.

An updated Lagrangian method with error estimation and adaptive remeshing for very large deformation elasticity problems

SUMMARYAccurate simulations of large deformation hyperelastic materials by the FEM is still a challenging problem. In a total Lagrangian formulation, even when using a very fine initial mesh, the simulation can break down due to severe mesh distortion. Error estimation and adaptive remeshing on the initial geometry are helpful and can provide more accurate solutions but are not sufficient to attain very large deformations. The updated Lagrangian formulation where the geometry is periodically updated is then preferred. However, it requires data transfer from the old mesh to the new one and this is a very delicate issue. In this paper, we present an updated Lagrangian formulation where the error is estimated and adaptive remeshing is performed in order to reach high level of deformations while controlling both the accuracy of the solution and mesh distortion. Special attention is given to data transfer methods and a very accurate cubic Lagrange projection method is introduced. A continuation method is used to automatically pilot the complete algorithm including load increase, error estimation, adaptive remeshing, and data transfer. A number of examples will be presented and analyzed. Copyright © 2014 John Wiley & Sons, Ltd.

A novel updated Lagrangian complementary energy-based formulation for the elastica problem: force-based finite element model

This paper addresses the development of a novel updated Lagrangian variational formulation and its associated finite element model for the geometrically nonlinear quasi-static analysis of cantilever beams. The formulation is based on an incremental complementary energy principle. The proposed finite element model only contains nodal bending moments as degrees of freedom. The model is used for the analysis of problems modeled by the so-called elastica theory. Numerical solutions satisfying all equilibrium equations in a strong sense can be obtained for arbitrarily large displacements and rotations. A Newton–Raphson method is adopted to trace the post-buckling response. Numerical results are presented and compared with those produced by the standard total Lagrangian two-node displacement-based finite element model.