Composite Finite Element Research Articles

Polytopal composite finite elements for modeling concrete fracture based on nonlocal damage models

The paper presents an assumed strain formulation over polygonal meshes to accurately evaluate the strain fields in nonlocal damage models. An assume strained technique based on the Hu-Washizu variational principle is employed to generate a new strain approximation instead of direct derivation from the basis functions and the displacement fields. The underlying idea embedded in arbitrary finite polygons is named as Polytopal composite finite elements (PCFEM). The PCFEM is accordingly applied within the framework of the nonlocal model of continuum damage mechanics to enhance the description of damage behaviours in which highly localized deformations must be captured accurately. This application is helpful to reduce the mesh-sensitivity and elaborate the process-zone of damage models. Several numerical examples are designed for various cases of fracture to discuss and validate the computational capability of the present method through comparison with published numerical results and experimental data from the literature.

Explicit topology optimization of nearly incompressible materials using polytopal composite elements

In this paper, an efficient explicit topology optimization approach to nearly incompressible materials was proposed. The crucial idea is based on a smooth projection of discrete moving morphable bars (MMBs) onto an analytical grid of polytopal composite finite elements (PCEs). The PCEs, which are built with piecewise linear shape functions and assumed strain fields, aim to mitigate the inherent volumetric locking issue and accurately model nearly incompressible materials. Unlike most existing techniques that use the mixed displacement-pressure formulation and conventional topology optimization, the current method is based on the pure displacement formulation and explicit topology optimization with only hundreds of design variables. Numerical results demonstrated that the proposed approach works stably and produces proper optimal designs of nearly incompressible structures without the mesh-dependence while filtering techniques or perimeter constraints are not required. The investigation promises the potential for solving a wide range of incompressible problems.

A unified formulation for composite quasi-2D finite elements based on global-local superposition

A unified theory to formulate a family of quasi-2D composite finite elements that capture through-the-thickness effects is proposed. The formulation is based on the superposition of global and local displacement fields and includes thermal effects, as a response to the claim of the scientific community. First are presented the fundamentals of the unified formulation: the kinematic assumptions; the imposition of continuity of displacements and transverse stresses, and of non-homogeneous boundary conditions; and the mathematical manipulations that render its numerical efficiency. Then, the new family of elements is assessed in problems involving different boundary conditions. Results are compared to analytical and highly refined 2D finite elements solutions, providing a benchmark of the elements capabilities. Mesh convergence is assessed, together with the influence of the global and local displacement fields. The numerical efficiency of the formulation is attested and an expansion for a 3D formulation is outlined.

Numerical modelling of 2D composite solids accounting for local fibre bending stiffness

This contribution deals with the development of a four-node membrane composite finite element to analyse two-dimensional (2D) fibre-reinforced solids taking into account the local bending stiffness of fibres. This plane composite element is a combination of a four-node membrane element with rotational degrees of freedom (DOFs), modelling a unit of the matrix domain, and two-node Timoshenko beam elements that represent fibres crossing each matrix element. The so-called ’Projected Fibre Approach’ is then considered to express the beam elements variables in terms of their corresponding matrix elements. Accordingly, the obtained system of equations size is equivalent to that of a non-reinforced medium which considerably reduces the computing cost. To assess the accuracy of the proposed plane element, three examples are presented and the obtained results are found to agree well with the reference solutions.

Stress, strain and electromechanical analyses of (RE)Ba2Cu3Ox conductors using three-dimensional/two-dimensional mixed-dimensional modeling: fabrication, cooling and tensile behavior

High temperature superconducting (HTS) conductors, represented by Rare Earth-Barium-Copper-Oxide (REBCO) conductors, are promising for high energy and high field superconducting applications. In practical applications, however, the HTS conductors experience different stresses and strains, including residual stresses due to thermal mismatch and tensile stresses due to Lorentz forces, resulting in some circumstances to a reduction in the load-carrying capacity as well as the risk of degradation in conductor critical current. In this study a mixed-dimensional high-aspect-ratio laminated composite finite element model for REBCO conductor is developed for stress and strain analyses in the processes of fabricating and cooling, as well as tensile testing. The model includes all the major constituent layers of a typical REBCO conductor and is experimentally validated. First, the thermal residual stresses and strains accumulated during the fabrication and cooling processes are analyzed by a multi-step modeling method that emulates the manufacturing process. Then, with the residual stresses and strains as initial stresses and strains, the mechanical behavior under a tensile load is studied. Lastly, a phenomenological critical current-strain model based on the Ekin power-law formula and the Weibull distribution function is combined with the mixed-dimensional conductor model to predict the strain dependence behavior of critical current in the reversible and irreversible degradation strain ranges. Simulation results show that the multi-step modeling is an effective method for stress and strain analyses of REBCO conductors during the fabrication and cooling processes and under and tensile loads. Compressive thermal residual stress generated on the REBCO layer during fabrication and cooling strongly affects the subsequent mechanical and current-carrying properties. Stress–strain curves generated by tensile loads are analyzed and experimentally validated at both the conductor and constituent-layer levels. Simulation results for the strain dependence of critical current are in good agreement with experiment data in both the reversible and irreversible degradation stages.

An adaptive composite discontinuous Galerkin method for elliptic problems on complicated domains with discontinuous coefficients

In this paper, we introduce the multi-region discontinuous Galerkin composite finite element method (MRDGCFEM) with hp-adaptivity for the discretization of second-order elliptic partial differential equations with discontinuous coefficients. This method allows for the approximation of problems posed on computational domains where the jumps in the diffusion coefficient form a micro-structure. Standard numerical methods could be used for such problems but the computational effort may be extremely high. Small enough elements to represent the underlying pattern in the diffusion coefficient have to be used. In contrast, the dimension of the underlying MRDGCFE space is independent of the complexity of the diffusion coefficient pattern. The key idea is that the jumps in the diffusion coefficient are no longer resolved by the mesh where the problem is solved; instead, the finite element basis (or shape) functions are adapted to the diffusion pattern allowing for much coarser meshes. In this paper, we employ hp-adaptivity on a series of test cases highlighting the practical application of the proposed numerical scheme.

Electro-vibration modeling and response of 3D skeletal frame configuration for energy harvesters

3D skeletal frame configuration is introduced as a means for overcoming deficiencies encountered by conventional 1D beam energy harvesters (EHs). A composite finite element model based on 3D beam kinematics is further developed and tailor-made for this purpose. A comprehensive electro-vibration analysis is conducted using a MATLAB code for the 3D skeletal frame EH as an enhanced application of the theory and compared to the corresponding 1D beam harvester. The results obtained by the theory is then validated against numerical responses based on active solid elements. It is shown that the novel 3D skeletal frame configuration offers substantial benefits over the conventional 1D beam including higher strain areas, multi-directionality, and effectively higher extracted power levels. These are actually among the most significant deficiencies by 1D beam topologies. The internal volume of the novel EH along with the surfaces of the tip mass can be utilized for electronic components and circuit boards.

Axons Embedded in a Tissue May Withstand Larger Deformations Than Isolated Axons Before Mechanoporation Occurs.

Diffuse axonal injury (DAI) is the pathological consequence of traumatic brain injury (TBI) that most of all requires a multiscale approach in order to be, first, understood and then possibly prevented. While in fact the mechanical insult usually happens at the head (or macro) level, the consequences affect structures at the cellular (or microlevel). The quest for axonal injury tolerances has so far been addressed both with experimental and computational approaches. On one hand, the experimental approach presents challenges connected to both temporal and spatial resolution in the identification of a clear axonal injury trigger after the application of a mechanical load. On the other hand, computational approaches usually consider axons as homogeneous entities and therefore are unable to make inferences about their viability, which is thought to depend on subcellular damages. Here, we propose a computational multiscale approach to investigate the onset of axonal injury in two typical experimental scenarios. We simulated single-cell and tissue stretch injury using a composite finite element axonal model in isolation and embedded in a matrix, respectively. Inferences on axonal damage are based on the comparison between axolemma strains and previously established mechanoporation thresholds. Our results show that, axons embedded in a tissue could withstand higher deformations than isolated axons before mechanoporation occurred and this is exacerbated by the increase in strain rate from 1/s to 10/s.

High-order composite finite element exact sequences based on tetrahedral–hexahedral–prismatic–pyramidal partitions

The combination of tetrahedral and hexahedral elements in a single conformal mesh requires pyramids or prisms to make the transition between triangular and quadrilateral faces. This paper presents high order exact sequences of finite element approximations in H1(Ω), H(curl,Ω), H(div,Ω), and L2(Ω) based on such kind of three dimensional mesh configurations. The approach is to consider composite polynomial approximations based on local partitions of the pyramids into two or four tetrahedra. The traces associated with triangular faces of these tetrahedral elements are constrained to match the quadrilateral shape functions on the quadrilateral face of the pyramid, in order to maintain conformity with shared neighboring hexahedron, or prism. Two classes of composite exact sequences are constructed, one using classic Nédélec spaces of first kind, and a second one formed by enriching these spaces with properly chosen higher order functions with vanishing traces. Projection-based interpolants satisfying the commuting diagram property are presented in a general form for each type of element. The interpolants are expressed as the sum of linearly independent contributions associated with vertices, edges, faces, and volume, according to the kind of traces appropriate to the space under consideration. Furthermore, we study applications to the mixed formulation of Darcy’s problems based on compatible pairs of approximations in {H(div,Ω),L2(Ω)} for such tetrahedral–hexahedral–prismatic–pyramidal meshes. An error analysis is outlined, showing same (optimal) orders of approximation in terms of the mesh size as one would obtain using purely hexahedral or purely tetrahedral partitions. Enhanced accuracy for potential and flux divergence variables is obtained when enriched space configurations are applied. The predicted convergence orders are verified for some test problems.

Polytopal composite finite elements

In recent years, polygonal and polyhedral finite elements have gained much interest from researchers as an alternative approach for discretizing complicated domains. Findings to enhance their performance are still desirable to solve a broad class of mechanics problems. In this study, we propose a new formulation based on arbitrary polytopes, which we name as Polytopal Composite finite Element (PCE). The key idea is to devise a polynomial projection of compatible strain fields through the least-squares approximation. For nearly incompressible problems, we design a pair of projection operators of volumetric and deviatoric strains resulting in stability of pressure solution. The present approach fulfills a patch test and satisfies the inf–sup stability. Through several numerical investigations, we show that the proposed method reaches the theoretical convergence rate and significantly improves the accuracy of polygonal element based solutions.

Error estimates for two-scale composite finite element approximations of parabolic equations with measure data in time for convex and nonconvex polygonal domains

In this exposition we study two-scale composite finite element approximations of parabolic problems with measure data in time for both convex and nonconvex polygonal domains. This research is motivated by the work of Hackbusch and Sauter [Numer. Math., 75 (1997) 447–472] on the composite finite element approximations of elliptic boundary value problems. The main features of the composite finite element method is that, it not only uses minimal dimension of the approximation space but also handle the domain boundary in a flexible and systematic manner, which is very advantageous for domains with complicated geometry. Both spatially semidiscrete and fully discrete approximations of the proposed method are analyzed. In the case of convex domains, we derive error estimate of order O(HLog˜1/2(H/h)+k1/2) in the L2(0,T;L2(Ω))-norm, where H and h denote the coarse-scale and fine-scale mesh size, respectively, and k is the time step. Further, an error estimate of order O(HsLog˜s/2(H/h)+k1/2), 1/2≤s≤1 is shown to hold in the L2(0,T;L2(Ω))-norm for nonconvex domains. Numerical experiment confirms the theoretical findings and reveals the potential of the composite finite element method.

Composite Finite Element Approximation for Parabolic Problems in Nonconvex Polygonal Domains

Abstract The purpose of this paper is to generalize known a priori error estimates of the composite finite element (CFE) approximations of elliptic problems in nonconvex polygonal domains to the time dependent parabolic problems. This is a new class of finite elements which was introduced by [W. Hackbusch and S. A. Sauter, Composite finite elements for the approximation of PDEs on domains with complicated micro-structures, Numer. Math. 75 1997, 4, 447–472] and subsequently modified by [M. Rech, S. A. Sauter and A. Smolianski, Two-scale composite finite element method for Dirichlet problems on complicated domains, Numer. Math. 102 2006, 4, 681–708] for the approximations of stationery problems on complicated domains. The basic idea of the CFE procedure is to work with fewer degrees of freedom by allowing finite element mesh to resolve the domain boundaries and to preserve the asymptotic order convergence on coarse-scale mesh. We analyze both semidiscrete and fully discrete CFE methods for parabolic problems in two-dimensional nonconvex polygonal domains and derive error estimates of order 𝒪 ( H s Log ^ ( H h ) s 2 ) {\mathcal{O}(H^{s}\widehat{\mathrm{Log}}{}^{\frac{s}{2}}(\frac{H}{h}))} and 𝒪 ( H 2 ⁢ s Log ^ ( H h ) s ) {\mathcal{O}(H^{2s}\widehat{\mathrm{Log}}{}^{s}(\frac{H}{h}))} in the L ∞ ⁢ ( H 1 ) {L^{\infty}(H^{1})} -norm and L ∞ ⁢ ( L 2 ) {L^{\infty}(L^{2})} -norm, respectively. Moreover, for homogeneous equations, error estimates are derived for nonsmooth initial data. Numerical results are presented to support the theoretical rates of convergence.

Deformation Criteria for Calculation of Composite Constructions

In this paper we consider the calculation of reinforcement and cross-sections of reinforced concrete structures: beams, columns, diaphragms of rigidity, etc. The article is divided into two parts. In the first part, the criterion for limiting deformations in given zones of the most stressed sections of composite systems is proposed as the basis for calculating reinforced concrete structures. The introduction of the appropriate limitations makes it possible to determine, from the equilibrium conditions of forces in a given section, one of the main geometric parameters of this section: height, width, cross-section of the working armature or the maximum possible load. An algorithm for solving the problem of selecting sections of a statically indeterminate system is shown. This approach is called the method of limited deformations (MLD). The method is illustrated by a number of examples. In the second part, the definition of points of nucleation of cracks and trajectories of their development is established by solving the problem of determining the deformed state of a composite under conditions of a plane stress state. Determination of the main deformations is performed by a modernized finite element method. The finite element calculation is based on specially developed composite elements describing the joint work of concrete and reinforcement. Two new (planar and spatial) composite finite elements are proposed. We show the scheme of formation of the general matrix of rigidity of a composite, as superposition of global matrixes of rigidity of a binder and armature.

Utilizing a Structural Mechanics Approach to Assess the Primary Effects of Injury Loads Onto the Axon and Its Components.

Diffuse axonal injury (DAI) occurs as a result of the transmission of rapid dynamic loads from the head to the brain and specifically to its neurons. Despite being one of the most common and most deleterious types of traumatic brain injury (TBI), the inherent cell injury mechanism is yet to be understood. Experimental observations have led to the formulation of different hypotheses, such as mechanoporation of the axolemma and microtubules (MTs) breakage. With the goal of singling out the mechanical aspect of DAI and to resolve the ambiguity behind its injury mechanism, a composite finite element (FE) model of a representative volume of an axon was developed. Once calibrated and validated against published experimental data, the axonal model was used to simulate injury scenarios. The resulting strain distributions along its components were then studied in dependence of strain rate and of typical modeling choices such as the applied MT constraints and tau proteins failure. Results show that oversimplifying the MT bundle kinematic entails a systematic attenuation (cf = 2.33) of the computed maximum MT strain. Nevertheless, altogether, results support the hypothesis of axolemma mechanoporation as a cell-injury trigger. Moreover, for the first time the interconnection between the axolemma and the MT bundle is shown to play a role in damage localization. The proposed FE axonal model is a valuable tool to understand the axonal injury mechanism from a mechanical perspective and could be used in turn for the definition of well-informed injury criteria at the tissue and organ level.

Composite finite element approximation for nonlinear parabolic problems in nonconvex polygonal domains

We study a new class of finite elements so‐called composite finite elements (CFEs), introduced earlier by Hackbusch and Sauter, Numer. Math., 1997; 75:447‐472, for the approximation of nonlinear parabolic equation in a nonconvex polygonal domain. A two‐scale CFE discretization is used for the space discretizations, where the coarse‐scale grid discretized the domain at an appropriate distance from the boundary and the fine‐scale grid is used to resolve the boundary. A continuous, piecewise linear CFE space is employed for the spatially semidiscrete finite element approximation and the temporal discretizations is based on modified linearized backward Euler scheme. We derive almost optimal‐order convergence in space and optimal order in time for the CFE method in the L∞(L2) norm. Numerical experiment is carried out for an L‐shaped domain to illustrate our theoretical findings.

General Framework for Modeling Multifunctional Metamaterial Beam Based on a Derived One-Dimensional Piezoelectric Composite Finite Element

AbstractPhononic crystals and metamaterials have been widely investigated over the last decade. In recent years, by integration with piezoelectric transducers, phononic/metamaterial-based piezoelec...

Multiscale smeared finite element model for mass transport in biological tissue: From blood vessels to cells and cellular organelles

One of the basic and vital processes in living organisms is mass exchange, which occurs on several levels: it goes from blood vessels to cells and organelles within cells. On that path, molecules, as oxygen, metabolic products, drugs, etc. Traverse different macro and micro environments – blood, extracellular/intracellular space, and interior of organelles; and also biological barriers such as walls of blood vessels and membranes of cells and organelles. Many aspects of this mass transport remain unknown, particularly the biophysical mechanisms governing drug delivery. The main research approach relies on laboratory and clinical investigations. In parallel, considerable efforts have been directed to develop computational tools for additional insight into the intricate process of mass exchange and transport. Along these lines, we have recently formulated a composite smeared finite element (CSFE) which is composed of the smeared continuum pressure and concentration fields of the capillary and lymphatic system, and of these fields within tissue. The element offers an elegant and simple procedure which opens up new lines of inquiry and can be applied to large systems such as organs and tumors models. Here, we extend this concept to a multiscale scheme which concurrently couples domains that span from large blood vessels, capillaries and lymph, to cell cytosol and further to organelles of nanometer size. These spatial physical domains are coupled by the appropriate connectivity elements representing biological barriers. The composite finite element has “degrees of freedom” which include pressures and concentrations of all compartments of the vessels-tissue assemblage. The overall model uses the standard, measurable material properties of the continuum biological environments and biological barriers. It can be considered as a framework into which we can incorporate various additional effects (such as electrical or biochemical) for transport through membranes or within cells. This concept and the developed FE software within our package PAK offers a computational tool that can be applied to whole-organ systems, while also including specific domains such as tumors. The solved examples demonstrate the accuracy of this model and its applicability to large biological systems.

Sensitivity analysis of a smart soft composite robotic finger design using geometrically nonlinear laminated composite finite beam elements

Soft robotics is an emerging research field that uses deformable materials to build compliant and adaptable systems, using simple integrated mechanisms, enabling biomimetic behavior. This paper presents a sensitivity analysis of a shape memory alloy (SMA)-driven smart soft composite (SSC) bending actuator that is used as an artificial finger in robots or prosthetics. This SSC structure is made of a flexible material, such as polydimethylsiloxane (PDMS), reinforced with some layers of glass fiber sheets, and actuated via SMA wires embedded in the flexible material. The SMA wires shrink when thermally activated, resulting in bending the three flexible hinges of the finger. Designing such smart soft structures involves optimizing large number of design parameters such as the flexible hinge length and thickness, the number of composite layers and their orientation, the number of SMA wires and their diameters. Hence an accurate and efficient computational tool is required to perform sensitivity analysis, and facilitate and accelerate the design process. The mathematical model should also account for geometric nonlinearity due to the large deformation and rotation of the structure in hand. A recently-developed accurate and efficient novel geometrically-nonlinear laminated composite beam finite element model is used in this work for analyzing the SSC robotic finger. The sensitivity analysis revealed that the effect of the flexible hinge thickness in the studied design is much more significant than the hinge length, and that the relationship between the fingertip rotation and the hinge thickness is nonlinear for the case of one glass fiber sheet and approaches linearity as the number of sheets increases to three.

Two-scale composite finite element method for parabolic problems with smooth and nonsmooth initial data

We consider a new class of finite elements, called composite finite elements, for the discretization of parabolic problems in a two-dimensional convex polygonal domain. More precisely, an effort has been made in this paper to extend two-scale composite finite element method for elliptic problems to parabolic problems. We analyze both semidiscrete and fully discrete composite finite element methods and derive convergence properties in the $$L^\infty (L^2)$$ and $$L^\infty (H^1)$$ norms for smooth data. Moreover, nonsmooth data error estimates in the $$L^\infty (L^2)$$ norm for positive time is established. Compared to the classical finite element method, the composite finite element method can be viewed as a coarse scale generalization which yields a minimal dimension of the approximation space and the asymptotic order of convergence is preserved on coarse scale meshes which do not resolve the boundary. Numerical results are presented to illustrate the theoretical rates of convergence.

Novel composite finite element model for piezoelectric energy harvesters based on 3D beam kinematics

In this study, piezoelectric smart structures are formulated using finite element method based on equivalent single layer theory for electromechanical domains, especially in energy harvesting. A model is derived where all displacement, strain, electric potential, and electric fields are decomposed into longitudinal and cross-sectional components and a finite element discretization is performed. All parameters are calculated for 3D active beam elements. Mechanical degrees of freedom are decoupled for both mechanical and electromechanical coupling matrices. Two local coordinate systems and a global one are introduced for this purpose. A method is presented and proved to transform element matrices between those systems. Natural frequencies of a cantilever harvester obtained from the theory are compared to those of ABAQUS model. Output voltage, current, RMS power, open and short circuit natural frequencies, and optimum resistive load are also calculated. Finally, the results of the present theory are compared to those presented in the literature.