Nonlinear Solid Mechanics Problems Research Articles

A physically and geometrically nonlinear formulation for isogeometric analysis of solids in boundary representation

AbstractThis contribution concerns a physically and geometrically nonlinear formulation for isogeometric analysis of solids in boundary representation. The parametrization of the proposed approach is inspired by the scaled boundary finite element method, where the geometry of the boundary is sufficient to define the entire solid surface. This suits perfectly the boundary representation modeling technique, which is commonly employed for the design of solids in CAD. In order to provide a boundary‐oriented approach, the solid surface is parametrized by a radial scaling parameter emanating from a scaling center and a parameter in the circumferential direction of the boundary. According to the idea of isogeometric analysis, NURBS basis functions define the geometry of the boundary and approximate the solution on the boundary. We also employ NURBS to approximate the solution in the scaling direction in the interior of the domain. This enables a straightforward treatment of nonlinear problems while preserving the exact geometry of the boundary. The approximation in scaling direction is in principle flexible and requires for coupling of adjacent patches inside the domain. The Galerkin method is employed for the solution in both parametric directions and a linearization is derived which is used within an iterative Newton‐Raphson scheme. We consider nonlinear problems in solid mechanics including elasto‐plastic material behavior and large deformations as well as complex geometries commonly employed in engineering applications. In this paper we study a numerical example of a complex geometry in order to evaluate the performance of the proposed approach.

A finite element reduced‐order model based on adaptive mesh refinement and artificial neural networks

SummaryIn this work, a reduced‐order model based on adaptive finite element meshes and a correction term obtained by using an artificial neural network (FAN‐ROM) is presented. The idea is to run a high‐fidelity simulation by using an adaptively refined finite element mesh and compare the results obtained with those of a coarse mesh finite element model. From this comparison, a correction forcing term can be computed for each training configuration. A model for the correction term is built by using an artificial neural network, and the final reduced‐order model is obtained by putting together the coarse mesh finite element model, plus the artificial neural network model for the correction forcing term.The methodology is applied to nonlinear solid mechanics problems, transient quasi‐incompressible flows, and a fluid‐structure interaction problem. The results of the numerical examples show that the FAN‐ROM is capable of improving the simulation results obtained in coarse finite element meshes at a reduced computational cost.

A novel analysis-prediction approach for geometrically nonlinear problems using group method of data handling

A novel analysis-prediction (ANP) approach for geometrically nonlinear problems of solid mechanics is for the first time proposed in this paper. The key concept of this approach is: (1) A part of equilibrium path is traced by numerical analysis; (2) Data of this part is then used for training predictive network; (3) Applying the trained network, the rest of the equilibrium path is simply traced by pure prediction without using any analysis. As an illustration for ANP approach, the analysis package is in this study established based on isogeometric shell analysis using the first-order shear deformation shell theory (FSDT). As the main advantage of the proposed approach, computational cost is significantly lower than that of the conventional approach based on pure numerical analysis. In addition, the predictive networks are built via group method of data handling (GMDH) known as a self-organizing deep learning method for time series forecasting problems without requirement of big data. Some numerical examples are provided to confirm the high accuracy and efficiency of the proposed approach. The approach not only could be applied to a wide range of computational mechanics problems in which nonlinear response occurs but also to other computational engineering fields.

A novel face-on-face contact method for nonlinear solid mechanics

The efficient solution of quasistatic contact problems in nonlinear solid mechanics continues to pose some challenges. Traditional node-to-segment methods are not guaranteed to pass the contact patch test, and exhibit non-smooth contact behavior in the presence of sliding. More recent mortar-based methods, which resolve contact interactions over local facet-pairs and typically feature an integral-form gap constraint, result in a more robust treatment of contact in the presence of geometric nonlinearities and large sliding. However, these methods usually require designation of one surface for the interpolation of contact quantities, and are therefore biased. Moreover, reliable iterative solution in the presence of contact constraints remains a challenge for all implicit contact methods. This work presents an unbiased face-on-face contact method using a median-plane methodology and an integral-form gap constraint. The method additionally features a novel subcycle iterative solution strategy that exhibits reliable and efficient convergence performance, even in the presence of large sliding and curved contacting surfaces. Performance of the method is demonstrated through a suite of nonlinear quasi-static contact problems.

A finite element formulation in boundary representation for the analysis of nonlinear problems in solid mechanics

The contribution is concerned with a numerical element formulation in boundary representation. It results in a polynomial element description with an arbitrary number of nodes on the boundary. Scaling the boundary description determines the interior domain. The scaling approach is adopted from the so-called scaled boundary finite element method (SBFEM), which is a semi-analytical formulation to analyze problems in linear elasticity. Within this method, the basic idea is to scale the boundary with respect to a scaling center. The boundary, which is denoted as circumferential direction, and the scaling direction span the parameter space. In the present approach, interpolations in scaling direction and circumferential direction are introduced. The interpolation in circumferential direction is independent of the scaling direction. The formulation is suitable to analyze problems in nonlinear solid mechanics. The displacement degrees of freedom are located at the nodes on the boundary and in the interior element domain. The degrees of freedom located at the interior domain are eliminated by static condensation, which leads to a polygonal finite element formulation with an arbitrary number of nodes on the boundary. The element formulation allows per definition for Voronoi meshes and quadtree mesh generation. Numerical examples give rise to the performance of the present approach in comparison to other polygonal element formulations, like the virtual element method (VEM). Some benchmark tests show the capability of the element formulation. A comparison to standard and mixed element formulations is presented. The present approach is perfectly suitable to model heterogeneous structures with inclusions and voids. It avoids also staircase approximation of curved boundaries.

A non-local damage approach for the boundary element method

The conventional (local) constitutive modelling of materials exhibiting strain softening behaviour is susceptive to a spurious mesh dependence caused by numerically induced strain localization. Also, for refined meshes, numerical instabilities may be verified, mainly if the simulations are performed by the boundary element method. An alternative to overcome such difficulties is the adoption of the so called non-local constitutive models. In these approaches, some internal variables of the constitutive model in a single point are averaged considering its values of the neighbouring points. In this paper, the implicit formulation of the boundary element method for physically non-linear problems in solid mechanics is used with a non-local isotropic damage model and a very simple averaging scheme, over internal cells, is introduced. It is shown that the analysis become more stable in comparison to the case of a local application of the same model and that the results recover the desired objectiveness to mesh refinement.

A finite element formulation in boundary representation for the analysis of nonlinear problems in solid mechanics

AbstractThe contribution is concerned with a numerical element formulation for the analysis of the nonlinear behavior in solid mechanics. It is based on the so‐called scaled boundary finite element method (SB‐FEM). The SB‐FEM is as a semi‐analytical formulation to analyze problems of linear elasticity. The basic idea is to scale the boundary with respect to a scaling center. Hence, the boundary denoted as circumferential direction and the scaling direction span the parameter space. The original SB‐FEM approach is based on an analytical ansatz for the displacement response preserving equilibrium in scaling direction. In the present approach, an interpolation in scaling direction is introduced to account for nonlinear problems. Lagrange interpolations with different polynomial order are therefore discussed. The interpolation in circumferential direction is independent of the scaling direction and is defined by independent Lagrange functions. The displacement degrees of freedom are located at the nodes on the boundary and in the interior element domain. The degrees of freedom located in the interior of the domain are eliminated by static condensation, which leads to a polygonal finite element formulation with an arbitrary number of nodes on the boundary. The element formulation allows a priori for Voronoi meshes and quadtree mesh generation. Numerical examples give rise to the performance of the present approach in comparison to other element formulations, like mixed formulations.

A stabilized mixed implicit Material Point Method for non-linear incompressible solid mechanics

In this work a stabilized mixed formulation for the solution of non-linear solid mechanics problems in nearly-incompressible conditions is presented. In order to deal with high material deformation, an implicit Material Point Method is chosen. Such choice allows avoiding the classical limitations of the Finite Element Method, e.g., element tangling and extreme mesh distortion. The proposed mixed formulation, with displacement and pressure as primary variables, is tested through classical benchmarks in solid and geo-mechanics where a Neo-Hookean, a J2 and a Mohr-Coulomb plastic law are employed. Further, the stabilized mixed formulation is compared with a displacement-based formulation to demonstrate how the proposed approach gets better results in terms of accuracy, not only when incompressible materials are simulated, but also in the case of compressible ones.

A Three-Node Triangular Element with Continuous Nodal Stress (Trig3-CNS) for Geometry Nonlinear Solid Mechanics Problems

This paper investigates the performance of the three-node triangular element with continuous nodal stress (Trig3-CNS) for geometry nonlinear solid mechanic problems. This Trig3-CNS element was recently proposed to improve accuracy of the finite element method (FEM). By synergizing the individual strengths of meshfree method and FEM, the Trig3-CNS element achieves higher accuracy and convergence rate. Furthermore, Trig3-CNS presents high tolerance to mesh distortion. Therefore, it is potentially useful for geometry nonlinear solid mechanics problems in which mesh distortion takes place. Compared with the traditional hybrid “FE-Meshfree” elements, Trig3-CNS naturally processes CNS without requiring any extra operation in post-processing. Numerical tests in the present work show that for geometry nonlinear analysis, the results of the Trig3-CNS element are better than the 3-node triangular element (Trig3) and 4-node isoparametric quadrilateral element (Quad4). In addition, the performance of Trig3-CNS is comparable to that of traditional hybrid “FE-Meshfree” elements.

Modification of the quadratic 10‐node tetrahedron for thin structures and stiff materials under large‐strain hyperelastic deformation

SummaryThe concept of energy‐sampling stabilization is used to develop a mean‐strain quadratic 10‐node tetrahedral element for the solution of geometrically nonlinear solid mechanics problems. The development parallels recent developments of a “composite” uniform‐strain 10‐node tetrahedron for applications to linear elasticity and nonlinear deformation. The technique relies on stabilization by energy sampling with a mean‐strain quadrature and proposes to choose the stabilization parameters as a quasi‐optimal solution to a set of linear elastic benchmark problems. The accuracy and convergence characteristics of the present formulation are tested on linear and nonlinear benchmarks and compare favorably with the capabilities of other mean‐strain and high‐performance tetrahedral and hexahedral elements for solids, thin‐walled structures (shells), and nearly incompressible structures.

A new polyhedral element for the analysis of hexahedral-dominant finite element models and its application to nonlinear solid mechanics problems

A hexahedral-dominant finite element mesh can be easily constructed by cutting regular hexahedral elements in a simple block with CAD surfaces representing outer surfaces of a geometric model. Polyhedral elements with straight edges but possibly non-planar faces are generated at the domain boundaries, while regular hexahedral elements remain in the interior region. Shape functions for polyhedral elements are derived from moving least square approximation based on a tetrahedral subdivision of polyhedral domains by a centroid-based subdivision technique. The polyhedral shape functions in this study have similar properties to conventional finite element shape functions in terms of continuity and completeness within elements, compatibility across inter-element boundaries and the Kronecker-delta property. Furthermore, the present approach using hexahedral-dominant meshes with polyhedral elements at domain boundaries is successfully applied to solve large deformation problems of hyperelastic and elastic–plastic materials.

A multilevel projection‐based model order reduction framework for nonlinear dynamic multiscale problems in structural and solid mechanics

SummaryA reduction/hyper reduction framework is presented for dramatically accelerating the solution of nonlinear dynamic multiscale problems in structural and solid mechanics. At each scale, the dimensionality of the governing equations is reduced using the method of snapshots for proper orthogonal decomposition, and computational efficiency is achieved for the evaluation of the nonlinear reduced‐order terms using a carefully designed configuration of the energy conserving sampling and weighting method. Periodic boundary conditions at the microscales are treated as linear multipoint constraints and reduced via projection onto the span of a basis formed from the singular value decomposition of Lagrange multiplier snapshots. Most importantly, information is efficiently transmitted between the scales without incurring high‐dimensional operations. In this proposed proper orthogonal decomposition–energy conserving sampling and weighting nonlinear model reduction framework, training is performed in two steps. First, a microscale hyper reduced‐order model is constructed in situ, or using a mesh coarsening strategy, in order to achieve significant speedups even in non‐parametric settings. Next, a classical offline–online training approach is performed to build a parametric hyper reduced‐order macroscale model, which completes the construction of a fully hyper reduced‐order parametric multiscale model capable of fast and accurate multiscale simulations. A notable feature of this computational framework is the minimization, at the macroscale level, of the cost of the offline training using the in situ or coarsely trained hyper reduced‐order microscale model to accelerate snapshot acquisition. The effectiveness of the proposed hyper reduction framework at accelerating the solution of nonlinear dynamic multiscale problems is demonstrated for two problems in structural and solid mechanics. Speedup factors as high as five orders of magnitude are shown to be achievable. Copyright © 2017 John Wiley & Sons, Ltd.

Mean‐strain 10‐node tetrahedron with energy‐sampling stabilization for nonlinear deformation

SummaryA mean‐strain 10‐node tetrahedral element is developed for the solution of geometrically nonlinear solid mechanics problems using the concept of energy‐sampling stabilization. A uniform‐strain tetrahedron for applications to linear elasticity was recently described. The formulation as extended here is able to solve large‐strain hyperelasticity. The present 10‐node tetrahedron is composed of several four‐node linear tetrahedral elements, four tetrahedra in the corners, and four tetrahedra that tile the central octahedron in three possible sets of four‐node tetrahedra, corresponding to three different choices for the internal diagonal. We formulate a mean‐strain element with stabilization energy evaluated on the four corner tetrahedra, which is shown to guarantee consistency and stability. The stabilization energy is expressed through a stored‐energy function, and contact with input parameters in the small‐strain regime is made. The neo‐Hookean model is used to formulate the stabilization energy. As for small‐strain elasticity, the stabilization parameters are determined by actual material properties and geometry of a tetrahedra without any user input. The numerical tests demonstrate that the present element performs well for solid, shell, and nearly incompressible structures. Copyright © 2016 John Wiley & Sons, Ltd.

Una formulación mixta estabilizada explícita para plasticidad con localización de deformaciones

This paper presents the application of stabilized mixed explicit strain/displacement formulation (MEX-FEM) [23,24] for solving non-linear plasticity problems in solid mechanics with strain localization. In order to use the same linear interpolation order for displacements and strains, the formulation uses the variational subscales method. Compared to the standard irreducible formulation, the proposed formulation yields improved strain and stress fields, and it is capable of addressing nearly incompressible situations. This work investigates the effects of the improved strain and stress fields in problems involving strain softening and localization leading to failure for the Mohr-Coulomb and Drucker Prager plasticity models. Numerical examples validate the ability of the proposed formulation to correctly predict failure mechanisms with localized patterns of strain, virtually free of mesh dependence and without using tracking algorithm.

УПАКОВКА И РАЗВЕРТЫВАНИЕ ВНУТРЕННИМ ДАВЛЕНИЕМ КРУПНОГАБАРИТНЫХ ОБОЛОЧЕЧНЫХ КОНСТРУКЦИЙ

The packaging of large composite shell structures (corrugation, a cylinder and a truncated cone) and their deployment by internal pressure loading are explored. It is believed that the medial surfaces of the constituent elements have involutes which coincide with them in a packed state. The corrugation consists of the ring components, the cylinder and cone consist of trapezoidal plane components. These components are made of carbon fiber with orthotropic or transversely isotropic elastic properties and stapled by joints. The joints do not perceive resistance to rotation about the tangent to the weld line. The contemplated structures perceive bending loads (unlike pneumatic ones) made of soft materials (fabrics, films). Geometrically nonlinear solid mechanics problems with the internal pressure loading are solved by using the engineering computing system ANSYS. The deployment pressure dependence on the shell material structure, shell thickness and amount of constituent elements are investigated. It is shown that the deployment pressure of the large shell is commensurate with the pressure of pneumatic structures of soft materials. It was found that the stresses in the corrugation shells can reach critical values but in the cylinder and the truncated cone the stresses are insignificant. The task formulation and its solution on the thermodynamic state of the injected gas under quasi-static internal pressure loading of the shell are suggested. It is shown that in the beginning of deployment the gas temperature will drop by about 50-80 degrees Celsius according to gas composition, and then its temperature is tending to increase to the injected gas temperature. These results enable to expand the choice of materials for the pneumatic products manufacturing including space applications design.

A three-dimensional hybrid smoothed finite element method (H-SFEM) for nonlinear solid mechanics problems

This paper presents a novel three-dimensional hybrid smoothed finite element method (H-SFEM) for solid mechanics problems. In 3D H-SFEM, the strain field is assumed to be the weighted average between compatible strains from the finite element method (FEM) and smoothed strains from the node-based smoothed FEM with a parameter α equipped into H-SFEM. By adjusting α, the upper and lower bound solutions in the strain energy norm and eigenfrequencies can always be obtained. The optimized α value in 3D H-SFEM using a tetrahedron mesh possesses a close-to-exact stiffness of the continuous system, and produces ultra-accurate solutions in terms of displacement, strain energy and eigenfrequencies in the linear and nonlinear problems. The novel domain-based selective scheme is proposed leading to a combined selective H-SFEM model that is immune from volumetric locking and hence works well for nearly incompressible materials. The proposed 3D H-SFEM is an innovative and unique numerical method with its distinct features, which has great potential in the successful application for solid mechanics problems.

Preface to special issue on nonlinear mechanics of solids

This special issue of the Archive of Applied Mechanics includes seven papers on mechanics of solids, which are selected from those orally presented at the sixth International Conference on NonlinearMechanics (ICNMVI), August, 12–15, 2013, Shanghai, China. The ICNM conference series was initiated in Shanghai in 1985 and traditionally covers a wide range of topical areas on nonlinear mechanics of solids and fluids. The seven papers were rigorously reviewed and subsequently revised by following the normal peer-review procedure of the journal. The problems investigated in the selected articles include the energy and entropy inflows into a continuum, the wavelet method for solving nonlinear solid mechanics problems, the method for constructing large-scale tensegrity structures, the opto-mechanical behavior of liquid crystal elastomers, the shape transition and stability of helical ribbon, the simulation of structural evolution of silicon nanowires, and the mechanical properties of locally oxidized graphene electrodes. The guest editor is grateful to the authors for their contributions. I also appreciate the reviewer’s constructive comments, which led to improvements in the technical quality and the presentation clarity. Finally, I wish to thank Professor Reinhold Kienzler, the editor of the journal, for inviting me to edit this special issue.

Mixed stabilized finite element methods in nonlinear solid mechanics. Part III: Compressible and incompressible plasticity

This paper presents the application of a stabilized mixed strain/displacement finite element formulation for the solution of nonlinear solid mechanics problems involving compressible and incompressible plasticity. The variational multiscale stabilization introduced allows the use of equal order interpolations in a consistent way. Such formulation presents two advantages when compared to the standard, displacement based, irreducible formulation: (a) it provides enhanced rate of convergence for the strain (and stress) field and (b) it is able to deal with incompressible situations. The first advantage also applies to the comparison with the mixed pressure/displacement formulation. The paper investigates the effect of the improved strain and stress fields in problems involving strain softening and localization leading to failure, using low order finite elements with continuous strain and displacement fields (P1P1 triangles or tetrahedra and Q1Q1 quadrilaterals, hexahedra, and triangular prisms) in conjunction with an associative frictional Drucker–Prager plastic model. The performance of the strain/displacement formulation under compressible and nearly incompressible deformation patterns is assessed and compared to a previously proposed pressure/displacement formulation. Benchmark numerical examples show the capacity of the mixed formulation to predict correctly failure mechanisms with localized patterns of strain, virtually free from any dependence of the mesh directional bias. No auxiliary crack tracking technique is necessary.

Modeling concrete like materials under sever dynamic pressures

The behavior of concrete, as well as that of all cohesive-frictional materials, is characterized by a strong dependence on hydrostatic pressure and strain rate sensitivity. The stress–strain response is nonlinear and, depending on the loading path, it can show compaction and dilatancy. Due to its internal constitution, the numerical simulation of this type of materials can be done with a great variety of approaches which are mainly differentiated by the modeling scales and the approximation methods used. Appropriate formulation and calibration of the models are necessary to accurately reproduce the physical phenomena involved in the simulated processes.A general elastic–viscoplastic model for concrete like materials under high strain rate dynamic loads that produce high confinement pressures, like those present in blast actions, is presented in this paper. The model developed is of phenomenological type and it is formulated within the framework of continuum thermodynamics for irreversible processes with internal variables and small strains. Non-associated plastic flow, cap function and hardening functions that depend on the loading path are proposed. The transition between compaction and dilatancy processes observed in triaxial compression experimental tests can be properly reproduced with the non-associated flow consideration and the proposed hardening functions. The cap function allows the appropriate simulation of the material volumetric response for high confinement pressures. The model formulation is general and it can be used for other cohesive-frictional materials.The numerical integration algorithm is implemented in a 2D finite element dynamic program that allows solving nonlinear problems of solid mechanics in small strains.For the model validation, static and dynamic tests with different loading paths and confinement levels for many types of concretes and mortars are analyzed. Numerical results are compared with experimental results obtaining a good description of the main characteristics of this type of materials response under high strain rate pressures.

Alya: Computational Solid Mechanics for Supercomputers

While solid mechanics codes are now conventional tools both in industry and research, the increasingly more exigent requirements of both sectors are fuelling the need for more computational power and more advanced algorithms. For obvious reasons, commercial codes are lagging behind academic codes often dedicated either to the implementation of one new technique, or the upscaling of current conventional codes to tackle massively large scale computational problems. Only in a few cases, both approaches have been followed simultaneously. In this article, a solid mechanics simulation strategy for parallel supercomputers based on a hybrid approach is presented. Hybrid parallelization exploits the thread-level parallelism of multicore architectures, combining MPI tasks with OpenMP threads. This paper describes the proposed strategy, programmed in Alya, a parallel multi-physics code. Hybrid parallelization is specially well suited for the current trend of supercomputers, namely large clusters of multicores. The strategy is assessed through transient non-linear solid mechanics problems, both for explicit and implicit schemes, running on thousands of cores. In order to demonstrate the flexibility of the proposed strategy under advance algorithmic evolution of computational mechanics, a non-local parallel overset meshes method (Chimera-like) is implemented and the conservation of the scalability is demonstrated.