Total Lagrangian Finite Element Research Articles

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.

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.

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.

Numerical study of large deflection of nanobeam according to the MCST and SE

In the current study, the large deflection of nanobeam according to the modified couple stress theory (MCST) with surface effects (SE) was investigated by applying finite element method (FEM). For modeling of the nanobeam, we applying the Timoshenko beam theory (TBT), and for the solution of the problem, we used a total lagrangian finite element method. Also, shear deformation and axial displacements of nanobeams have been considered in this model. The FEM has been used for the solution of the problem. With the present model, small to large deflection of nanobeam with arbitrary loadings and boundary conditions can be studied. The numerical results for the small deflection of nanobeam achieved by the present model are well matched with results reported in the previous researches. Effects of the ratio of material length scale parameter to thickness on the large deflection of nanobeams have been investigated. Also, the influence of considering different beam constitutive theories in the large deflection analysis of nanobeams has been examined. The results achieved in this study show the relative importance of considering surfaces and size effects in the analysis of nanobeam.

Improved friction model applied to plane sliding connections by a large deformation FEM formulation

Friction is an important source of dissipation in dynamical systems. Properly considering it in the numerical model is fundamental to obtain stable and representative responses in structures and mechanisms. This is especially significant for the well-known Coulomb model due to discontinuity in force when stick-slip transition occurs. In this work an improved friction force model is proposed to smooth the force transition at null velocity, with an additional parameter obtained from the own system state. The improved model is employed in sliding connections of plane frames finite elements. A total Lagrangian Finite Element Method (FEM) formulation based on a positional description of the motion is employed. Using a variational principle, frictional dissipation is added to the total mechanical energy to develop the equations of motion. The resulting nonlinear equations are solved by the Newton-Raphson method accounting for the friction force update in the iterative process. Examples are presented to show the formulation effectiveness and possibilities in simulating dynamical systems that present the stick-slip effect.

Optimal design of geometrically nonlinear structures using topology optimization

This study deals with comparing the stiffness design of geometrically nonlinear structures and linear structures using topology optimization. Bi-directional Evolutionary Structures Optimization (BESO) is used for the design process. The geometrically nonlinear behavior of the structures is analyzed using a total Lagrangian finite element formulation and the equilibrium is achieved by Newton-Raphson iterative scheme. The topology optimization of linear and nonlinear modeling is performed. The sensitivity of the objective function is found with the adjoint method and the optimization problem is solved using BESO’s update method. Objective function of complementary work is evaluated. A special technique, the continuation method, is applied to eliminate the instability of nonlinear structure optimization. ANSYS APDL is also used to do FEA of optimal topology to verify the effectiveness of geometrically nonlinear modelling. The results show that differences in stiffness of structures optimized using linear and nonlinear modelling are generally small but it can be large in some cases, especially structure highly involving buckling behaviour.

Growth mechanics of the viscoelastic membranes

Living membranes can grow as a response to their environment, and they are also known as viscoelastic materials. However, the growth phenomenon and the viscoelastic behavior of these membranes have been studied separately up to now. The main goal of this paper is to develop a general formulation that can account for growth and viscoelastic behaviors in membranes, simultaneously. Therefore, we propose a formulation for growing quasi-linear viscoelastic membranes. The growth model begins with a multiplicative decomposition of the gradient deformation tensor into elastic and growth tensor, where the growth of the membrane is considered as a single scalar-valued parameter. Furthermore, the quasi-linear viscoelasticity theory is utilized in the elastic configuration as additive decomposition of the second Piola–Kirchhoff stress tensor. To solve the evolution equations for both growth and quasi-linear viscoelastic relations, two implicit second order accuracy integration methods are employed. Moreover, strain-driven and stress-driven growth phenomena are accounted for based on a Total Lagrangian (TL) finite element formulation. The developed formulations can capture growth phenomena and viscoelastic properties of the membranes, individually, and it also enables to model co-existence of growth and viscoelasticity. Eventually, to evaluate the applicability of the developed formulation, several examples are investigated and compared to those available in the literature.

Large Amplitude Free Vibration Analysis of Nanobeams Based on Modified Couple Stress Theory

In this study, the large amplitude free vibration of nanobeams based on the modified couple stress theory was developed by using Total Lagrangian finite element formulation. In this study, Timoshenko beam theory has been used in free vibration analysis of nanobeams. Minimal kinematic assumptions have been used to model nanobeams. With this model, free vibration of nanobeams with small to large amplitude and with arbitrary boundary conditions can be analyzed. The numerical results obtained for free vibration based on the modified couple stress theory with small amplitude and the results obtained for free vibration with large amplitude without considering the modified couple stress theory are in good agreement with the similar results reported in previous research. Effects of the dimensionless length scale parameter, slenderness ratio, vibration amplitude and different boundary conditions on the nonlinear frequency ratio of nanobeams have been investigated. The results show the importance of considering nonlinear and size effects in the free vibration analysis of nanobeams with large amplitude.

Plastic work constrained elastoplastic topology optimization

AbstractAn elastoplastic topology optimization framework for limiting plastic work generation while maximizing stiffness is presented. The kinematics and constitutive model are based on finite strain linear isotropic hardening plasticity, and the balance laws are solved using a total Lagrangian finite element formulation. Aggregation of the specific plastic work combined with an adaptive normalization scheme efficiently constrains the maximum specific plastic work. The optimization problem is regularized using an augmented partial differential equation filter, and is solved by the method of moving asymptotes where path‐dependent sensitivities are derived using the adjoint method. The numerical examples show a clear dependence on the optimized maximum stiffness structures for different levels of constrained specific plastic work. It is also shown that due to the history dependency of the plasticity, the load path significantly influences the structural performance and optimized topology.

Large deformation analysis of fully incompressible hyperelastic curved beams

The aim of this contribution is to develop a nonlinear formulation for modelling the large deformation of elastic curved beams made of fully incompressible hyperelastic materials. The basic idea is to apply the incompressibility constraint besides the plane stress assumption in the directions perpendicular to the centreline of the beam. Accordingly, a system of three nonlinear algebraic equations for the hydrostatic pressure as well as normal strain components in the beam cross section are obtained. By solving the system of equations, either analytically or numerically, it is possible to propose constitutive equations for the force and moment resultants present in the formulation. The main advantage of this strategy is that it allows for using three-dimensional incompressible constitutive equations. Due to highly nonlinear nature of the differential equations, a Total Lagrangian finite element formulation is developed. Performance and accuracy of the formulation are investigated through several numerical examples.

Topology optimization and 3D printing of large deformation compliant mechanisms for straining biological tissues

This paper presents a synthesis approach in a density-based topology optimization setting to design large deformation compliant mechanisms for inducing desired strains in biological tissues. The modelling is based on geometrical nonlinearity together with a suitably chosen hypereleastic material model, wherein the mechanical equilibrium equations are solved using the total Lagrangian finite element formulation. An objective based on least-square error with respect to target strains is formulated and minimized with the given set of constraints and the appropriate surroundings of the tissues. To circumvent numerical instabilities arising due to large deformation in low stiffness design regions during topology optimization, a strain-energy based interpolation scheme is employed. The approach uses an extended robust formulation i.e. the eroded, intermediate and dilated projections for the design description as well as variation in tissue stiffness. Efficacy of the synthesis approach is demonstrated by designing various compliant mechanisms for providing different target strains in biological tissue constructs. Optimized compliant mechanisms are 3D-printed and their performances are recorded in a simplified experiment and compared with simulation results obtained by a commercial software.

A selective smoothed finite element method with visco‐hyperelastic constitutive model for analysis of biomechanical responses of brain tissues

AbstractBrain tissues are known for exhibiting complex nonlinear and time‐dependent properties, which require visco‐hyperelastic constitutive models for proper simulation. In this paper, a Total Lagrangian Explicit Selective Smoothed Finite Element Method (Selective S‐FEM) is formulated to analyze the dynamic behavior of incompressible brain tissues undergoing extremely large deformation. The proposed Selective S‐FEM deals with three‐dimensional problems using four‐node tetrahedron elements that can be automatically generated for geometrically complex soft tissues. It consists of the three key ingredients. (i) A visco‐hyperelastic constitutive model is developed within the framework of S‐FEM in the first time, allowing adequate modeling of the dynamic brain tissue behavior. (ii) Selective S‐FEM strategy is used for overcome the mesh distortion and the volumetric locking that often occurs in soft tissues. (iii) Total Lagrangian formulation is used in an explicit algorithm allowing rigorous simulation of extreme large deformation. (iv) A combined implementation of Selective S‐FEM with the visco‐hyperelastic constitutive model for dynamic simulations. The shear deformation is calculated by Face/Edge‐based S‐FEM, and the volume deformation is calculated by NS‐FEM. Numerical experiments show that Selective S‐FEM is a robust solver with good accuracy, and excellent ability to reduce element distortion effects in simulate time‐dependence behavior of bio‐tissues.

Multi-time scale based modeling of piezoelectric materials coupling transient electrical and dynamic fields with finite deformation damage

This paper develops a high performance scalable computational model for simulating coupled transient electric and finite deformation dynamic fields with widely discrepant frequencies, governing the behavior of piezoelectric materials in the finite element framework. The nonlinear piezoelectric constitutive laws are formulated in the context of continuum damage models (CDM) at finite deformation. A fully coupled, total Lagrangian finite element formulation is developed for modeling the electric and mechanical fields in a staggered way. To address the computational challenge in modeling piezoelectric devices with large discrepancy in the frequenlightsghtscies of the electrical signal and mechanical vibrations, a wavelet transformation induced multi-time scaling (WATMUS) algorithm is developed. The WATMUS method projects the field with higher frequency onto the lower frequency problem through the use of orthogonal wavelet transformation. No assumption of periodic response is needed in the WATMUS scheme. While being accurate, the WATMUS algorithm significantly enhances the computational efficiency in comparison with single time scale integration methods. The accuracy and efficiency of the WATMUS algorithm are validated through piezoelectric applications.

Alternative active nonlinear total Lagrangian truss finite element applied to the analysis of cable nets and long span suspension bridges

An alternative geometrically nonlinear total Lagrangian finite element is presented and applied to solve cable, cable nets and a very long suspended bridge in both three and two-dimensional spaces from its setting-up through its response to earthquake. It includes dynamics, pseudo-dynamics regularization, elastic actuators and automatic stress calibration. Dynamics and pseudo-dynamics are used to perform transient structural analysis and the setting-up of very unstable structures. Elastic actuators allow pre-stressing structural members for the iterative structural design and cables natural length definition. Automatic stress calibration comprises continuous cables in complicated structures without sliding contact devices. The formulation is applied to model main cables of suspended bridges passing through saddle points. Inertial terms are introduced by an alternative mathematical way. Two simple examples are used to validate all aspects of the proposed formulation. Finally, a representative application is performed, i.e., the numerical design and analysis of a very long span suspension bridge by the proposed strategy.

A total Lagrangian position-based finite element formulation for free-surface incompressible flows

In this work, we propose a position-based finite element formulation for incompressible Newtonian flows under total Lagrangian description. Such formulation is different from the traditional finite element approach used in fluid dynamics by using current nodal positions as main variable instead of nodal velocities. The variational form of the governing equations is derived by applying the stationary total mechanical energy principle written in terms of current positions. To introduce full incompressibility, we use equal-order mixed finite elements combined to a pressure stabilizing Petrov-Galerkin technique, circumventing Ladyzhenskaya-Babuška-Brezzi conditions. This leads to a method directly applicable to finite strain incompressible flow problems that, if combined to re-meshing techniques, is capable of simulating more complex problems, resulting into a suitable tool for free-surface flows analysis in general. The efficiency and robustness of the proposed approach is demonstrated with numerical studies and comparison of obtained results to numerical and experimental data from literature.

Computational modeling of finite deformation piezoelectric material behavior coupling transient electrical and mechanical fields

This paper develops a finite element formulation and computational model for coupling electric and finite strain dynamic fields with widely discrepant frequencies, governing the behavior of piezoelectric materials. The piezoelectric materials are defined by time-dependent nonlinear constitutive laws. A fully coupled, total Lagrangian finite element formulation is developed for modeling the electric and mechanical fields. A challenge in computational modeling of piezoelectric devices arise due to large discrepancy in the frequencies of the electrical signal and mechanical vibrations, especially when a large number of mechanical cycles need to be simulated. A wavelet transformation induced multi-time scaling (WATMUS) algorithm is developed for dynamic piezoelectric simulations. The WATMUS algorithm projects the high frequency electric fields on to the lower frequency of displacement and velocity fields, on which time integration is performed. The method significantly enhances the computational efficiency in comparison with single time scale integration methods. The accuracy and efficiency of the WATMUS algorithm are validated through piezo-electric applications.

Topology optimization of finite strain viscoplastic systems under transient loads

SummaryA transient finite strain viscoplastic model is implemented in a gradient‐based topology optimization framework to design impact mitigating structures. The model's kinematics relies on the multiplicative split of the deformation gradient, and the constitutive response is based on isotropic hardening viscoplasticity. To solve the mechanical balance laws, the implicit Newmark‐beta method is used together with a total Lagrangian finite element formulation. The optimization problem is regularized using a partial differential equation filter and solved using the method of moving asymptotes. Sensitivities required to solve the optimization problem are derived using the adjoint method. To demonstrate the capability of the algorithm, several protective systems are designed, in which the absorbed viscoplastic energy is maximized. The numerical examples demonstrate that transient finite strain viscoplastic effects can successfully be combined with topology optimization.

An explicit algorithm for geometrically nonlinear transient analysis of spatial beams using a corotational total Lagrangian finite element formulation

An explicit method for nonlinear transient dynamic analysis of spatial beams with finite rotations using a corotational total Lagrangian finite element formulation is presented. The kinematics of the beam element is described in the current element coordinate system constructed in the current configuration of the beam element. The element deformation and inertia nodal forces are derived by the virtual work principle, the d'Alembert principle, and the consistent linearization of the geometrically nonlinear beam theory. A nodal rotation vector is used to represent the finite rotation of a base coordinate system rigidly attached to each node of the discretized structure. A numerical procedure of explicit method is proposed for the solution of the nonlinear equations of motion. The standard central difference method is applied to the incremental displacement vector and the incremental rotation vector, and the time derivatives of displacement vector and rotation vector. The nodal orientations are updated by the incremental nodal rotation vectors. The values of nodal rotation vectors are reset to zero in the current configuration.In order to assess the efficiency and the accuracy of the proposed method, numerical examples are studied and compared with the results obtained using the implicit method based on the Newmark method.

Topology optimization of geometrically nonlinear structures using an evolutionary optimization method

ABSTRACTThe topology optimization using isolines/isosurfaces and extended finite element method (Iso-XFEM) is an evolutionary optimization method developed in previous studies to enable the generation of high-resolution topology optimized designs suitable for additive manufacture. Conventional approaches for topology optimization require additional post-processing after optimization to generate a manufacturable topology with clearly defined smooth boundaries. Iso-XFEM aims to eliminate this time-consuming post-processing stage by defining the boundaries using isovalues of a structural performance criterion and an extended finite element method (XFEM) scheme. In this article, the Iso-XFEM method is further developed to enable the topology optimization of geometrically nonlinear structures undergoing large deformations. This is achieved by implementing a total Lagrangian finite element formulation and defining a structural performance criterion appropriate for the objective function of the optimization problem. The Iso-XFEM solutions for geometrically nonlinear test cases implementing linear and nonlinear modelling are compared, and the suitability of nonlinear modelling for the topology optimization of geometrically nonlinear structures is investigated.

Nonlinear static analysis of functionally graded porous beams under thermal effect

This paper deals with the nonlinear static deflections of functionally graded (FG) porous under thermal effect. Material properties vary in both position-dependent and temperature-dependent. The considered nonlinear problem is solved by using Total Lagrangian finite element method within two-dimensional (2-D) continuum model in the Newton-Raphson iteration method. In numerical examples, the effects of material distribution, porosity parameters, temperature rising on the nonlinear large deflections of FG beams are presented and discussed with porosity effects. Also, the effects of the different porosity models on the FG beams are investigated in temperature rising.