Three-dimensional Finite Element Formulation Research Articles

A three-dimensional finite element formulation coupling electrochemistry and solid mechanics on resolved microstructures of all-solid-state lithium-ion batteries

In this paper we present an all-solid-state lithium-ion battery (ASSB) model that considers electrochemistry and solid mechanics in a fully coupled manner. The model spatially resolves the three-dimensional microstructure of an ASSB cell and is based on non-linear continuum mechanics. The coupling of electrochemistry and solid mechanics is incorporated via lithiation-dependent volumetric changes of the active materials and the consistent formulation of the electrochemical governing equations on deformed geometries resulting in changed percolation paths. As the volumetric changes due to (de-)lithiation are reported to be large for several active materials, we introduce this effect using a multiplicative split of the deformation gradient. To also allow for large deformations in the mass conservation equation, an Arbitrary Lagrangian–Eulerian formulation is employed. Furthermore, we show the central steps to deduce the finite element formulation of our model as well as the adopted monolithic solution procedure to solve this strongly coupled non-linear problem. The linear solver used for large problem setups is adjusted from the advanced, physics-oriented preconditioning technique published in our previous work. In the numerical examples, we investigate three different geometries with up to about 1.3 million degrees of freedom and thereby prove that our approach is applicable to geometrically large and complex scenarios. Moreover, we show that our approach consistently introduces the mass and charge conservation on deformed meshes. Furthermore, the numerical examples demonstrate the high relevance to incorporate current collectors into the ASSB cell simulations. Finally, we show that solid mechanics and the coupling effects have indeed a large impact on the ASSB cell performance and their incorporation is therefore vital for precise results and predictions.

Multiple equilibrium states of a curved-sided hexagram: Part I—stability of states

The stability of the multiple equilibrium states of a hexagram ring with six curved sides is investigated. Each of the six segments is a rod having the same length and uniform natural curvature. These rods are bent uniformly in the plane of the hexagram into equal arcs of 120o or 240o and joined at a cusp where their ends meet to form a 1-loop planar ring. The 1-loop rings formed from 120o or 240o arcs are inversions of one another and they, in turn, can be folded into a 3-loop straight line configuration or a 3-loop ring with each loop in an “8” shape. Each of these four equilibrium states has a uniform bending moment. Two additional intriguing planar shapes, 6-circle hexagrams, with equilibrium states that are also uniform bending, are identified and analyzed for stability. Stability is lost when the natural curvature falls outside the upper and lower limits in the form of a bifurcation mode involving coupled out-of-plane deflection and torsion of the rod segments. Conditions for stability, or lack thereof, depend on the geometry of the rod cross-section as well as its natural curvature. Rods with circular and rectangular cross-sections will be analyzed using a specialized form of Kirchhoff rod theory, and properties will be detailed such that all four of the states of interest are mutually stable. Experimental demonstrations of the various states and their stability are presented. Part II presents numerical simulations of transitions between states using both rod theory and a three-dimensional finite element formulation, includes confirmation of the stability limits established in Part I, and presents additional experimental demonstrations and verifications.

Three-dimensional Non-linear Finite Element Analysis of Impact Response and Damage in Laminated Composite Shells

The impact response and the impact-induced damage in curved composite laminates subjected to transverse impact by a metallic impactor are studied using three-dimensional non-linear transient dynamic finite element formulation. A layered version of isoparametric eight-noded hexahedral element with incompatible modes is developed which incorporates geometrical non-linearity based on total Langragian approach. The non-linearity of both strain displacement relation and contact loading are simultaneously solved using Newton-Raphson incrementaliterative method. Impact-induced damages (matrix cracking and delamination) are predicted using appropriate three-dimensional stress-based failure criteria. Some example problems of graphite/epoxy laminated cylindrical shells with variation of important parameters such as impactor velocity, shell curvature, laminate dimension and fibre orientation of plies are solved and the influence of geometrical non-linear effect on both impact response and resulting damages is demonstrated.

Convection-Enhanced Delivery In Silico Study for Brain Cancer Treatment.

Brain cancer therapy remains a formidable challenge in oncology. Convection-enhanced delivery (CED) is an innovative and promising local drug delivery method for the treatment of brain cancer, overcoming the challenges of the systemic delivery of drugs to the brain. To improve our understanding about CED efficacy and drug transport, we present an in silico methodology for brain cancer CED treatment simulation. To achieve this, a three-dimensional finite element formulation is utilized which employs a brain model representation from clinical imaging data and is used to predict the drug deposition in CED regimes. The model encompasses biofluid dynamics and the transport of drugs in the brain parenchyma. Drug distribution is studied under various patho-physiological conditions of the tumor, in terms of tumor vessel wall pore size and tumor tissue hydraulic conductivity as well as for drugs of various sizes, spanning from small molecules to nanoparticles. Through a parametric study, our contribution reports the impact of the size of the vascular wall pores and that of the therapeutic agent on drug distribution during and after CED. The in silico findings provide useful insights of the spatio-temporal distribution and average drug concentration in the tumor towards an effective treatment of brain cancer.

Numerical Investigation of High-Order Solid Finite Elements for Anisotropic Finite Strain Problems

In this paper, a hierarchic high-order three-dimensional finite element formulation is studied for hyperelastic and anisotropic elastoplastic problems at finite strains. The element formulation allows for anisotropic ansatz spaces supporting efficient discretizations of beam-, plate-, and shell-like structures. Several benchmark examples are investigated and the results of the high-order formulation are compared to analytic solutions and different mixed finite element formulations. Special emphasis will be placed on locking effects, robustness with respect to high aspect ratios and element distortion as well as anisotropies related to the material model. Furthermore, the interplay between the chosen ansatz space for the displacement field and mapping function in the context of geometrically nonlinear problems are studied.

Spline-based space-time finite element approach for fluid-structure interaction problems with a focus on fully enclosed domains

Non-Uniform Rational B-Spline (NURBS) surfaces are commonly used within Computer-Aided Design (CAD) tools to represent geometric objects. When using isogeometric analysis (IGA), it is possible to use such NURBS geometries for numerical analysis directly. Analyzing fluid flows, however, requires complex three-dimensional geometries to represent flow domains. Defining a parametrization of such volumetric domains using NURBS can be challenging and is still an ongoing topic in the IGA community.With the recently developed NURBS-enhanced finite element method (NEFEM), the favorable geometric characteristics of NURBS are used within a standard finite element method. This is achieved by enhancing the elements touching the boundary by using the NURBS geometry itself. In the current work, a new variation of NEFEM is introduced, which is suitable for three-dimensional space-time finite element formulations. The proposed method makes use of a new mapping which results in a non-Cartesian formulation suitable for fluid-structure interaction (FSI).This is demonstrated by combining the method with an IGA formulation in a strongly-coupled partitioned framework for solving FSI problems. The framework yields a fully spline-based representation of the fluid-structure interface through a single NURBS.The coupling conditions at the fluid-structure interface are enforced through a Robin-Neumann type coupling scheme. This scheme is particularly useful when considering incompressible fluids in fully Dirichlet-bounded and curved problems, as it satisfies the incompressibility constraint on the fluid for each step within the coupling procedure.The accuracy and performance of the introduced spline-based space-time finite element approach and its use within the proposed coupled FSI framework are demonstrated using a series of two- and three-dimensional benchmark problems.

Finite element formulation to study thermal stresses in nanoencapsulated phase change materials for energy storage

Nanoencapsulated phase change materials (nePCMs) – which are composed of a core with a phase change material and of a shell that envelopes the core – are currently under research for heat storage applications. Mechanically, one problem encountered in the synthesis of nePCMs is the failure of the shell due to thermal stresses during heating/cooling cycles. Thus, a compromise between shell and core volumes must be found to guarantee both mechanical reliability and heat storage capacity. At present, this compromise is commonly achieved by trial and error experiments or by using simple analytical solutions. On this ground, the current work presents a thermodynamically consistent and three-dimensional finite element (FE) formulation considering both solid and liquid phases to study thermal stresses in nePCMs. Despite the fact that there are several phase change FE formulations in the literature, the main novelty of the present work is its monolithic coupling – no staggered approaches are required – between thermal and mechanical fields. Then, the FE formulation is implemented in a computational code and it is validated against one-dimensional analytical solutions. Finally, the FE model is used to perform a thermal stress analysis for different nePCM geometries and materials to predict their mechanical failure by using Rankine’s criterion.

A finite element–based assessment of free vibration behaviour of circular and annular magneto-electro-elastic plates using higher order shear deformation theory

In this article, the free vibration behaviour of circular and annular magneto-electro-elastic plates has been investigated under the framework of higher order shear deformation theory. The three-dimensional finite element formulation has been derived with the aid of Hamilton’s principle by taking into account the coupling between elastic, electric and magnetic properties. The equations of motion are solved using condensation technique. Furthermore, the credibility of proposed finite element formulation has been validated using COMSOL software and also by comparing the results with previously published articles. Special attention has also been paid to assess the influence of parameters such as coupling effect, stacking sequences and inner-to-outer diameter ratio. The numerical results reveal that the coupled natural frequencies of the annular magneto-electro-elastic plates vary significantly with the circular hole dimensions incorporated. The circular and annular plates are considered as one of the prominent structural components in various engineering and industrial application. Therefore, the proposed finite element formulation and the results presented in this article can serve as benchmark solutions for the design and analysis of smart sensors and actuators.

Three-dimensional explicit finite element formulation for shear localization with global tracking of embedded weak discontinuities

Shear localization is a damage mechanism characterized by the concentration of plastic deformation within thin bands of materials, often observed as a precursor to failure in metals. In this paper, we present a three-dimensional explicit finite element formulation with embedded weak discontinuities for the treatment of shear localization under dynamic loading conditions. In this formulation, the onset of localization is detected via a material stability analysis suitable for rate-sensitive materials, and the continuity of propagating shear bands is ensured using a global tracking strategy involving a heat-conduction type boundary value problem, solved for a scalar level set function over the global domain of the problem. In addition, we propose a modified quadrature rule that is used to compute the contribution of an individual element to the global finite element arrays, taking into account the position of the embedded shear band within that element. The mechanical threshold stress (MTS) model, modified to account for dynamic recrystallization, is adopted as the constitutive model for rate- and temperature-sensitive metallic materials, and the constitutive equations are formulated in a corotational setting. The algorithmic implementation of the integrated finite element framework, including the global tracking strategy, is described in detail. Subsequently, a simple example problem involving shear band formation under uni-axial tension is presented, to illustrate the advantages of the proposed computational framework over conventional finite element techniques, in terms of convergence and mesh insensitivity of their numerical results. In addition, to demonstrate the capabilities of the proposed computational framework in more realistic problems, numerical results are compared to data from split-Hopkinson pressure bar dynamic experiments, conducted on two stainless steel samples with different geometric designs. In each problem presented, we study the effect of the mesh size on the numerical solution, including global results such as the load–displacement response, and local quantities such as material state variables at individual quadrature points. These numerical studies show that the geometry and total volume of the fully-developed embedded shear band are independent of the mesh size.

EFFECT OF TRANSIENT THERMAL GRADIENT ON STRESSES IN COMPOSITE BRIDGES

Composite steel-concrete bridges experience higher thermally induced stresses than their concrete and steel cousins. These thermal stresses which can result from support restraints, debris accumulating in expansion joints, or from non-uniform thermal gradients, can lead to significant damage in the concrete deck. Conventional heat transfer theory in solids, in three-dimensional finite element formulation, is used to perform a sequentially coupled thermal-stress analysis in a selected single-span case study bridge. Actual environmental boundary conditions for a selected geographical region are used to develop the thermal profile. The vertical thermal gradient is shown to be largely non-linear as opposed to existing models, such as AASHTO. The thermally induced tensile stresses in the concrete are shown to be significant compared to service load stresses and constitute 60% of the tensile strength of the concrete deck.

Vibration control of skew magneto-electro-elastic plates using active constrained layer damping

In this article, the effect of active constrained layer damping (ACLD) on the linear frequency response characteristics of skew magneto-electro-elastic (SMEE) plates has been evaluated. The constraining and the constrained layer of ACLD treatment are considered to be made of 1–3 piezoelectric composite (PZC) and viscoelastic layer, respectively. The advantage of incorporating smart materials such as PZC in attenuating the vibrations of the SMEE plates is thoroughly investigated. In this regard, a three-dimensional finite element (FE) formulation is derived with the aid of the principle of virtual work considering the coupling between magnetic, electric and elastic fields. The SMEE plate kinematics is governed by layer-wise shear deformation theory. The Complex modulus approach is adopted in order to model the constrained layer of the ACLD patch. A special emphasis is placed on investigating the effect of various skew angles on the frequency response of SMEE plates with ACLD treatment. Meanwhile, an exhaustive parametric study is carried out to analyze the influence of control gain, stacking sequence, patch position, fiber orientation angle of PZC, aspect ratio, and coupling effects. The results confirm that these parameters in association with ACLD treatment and skew angle significantly affect the damping characteristics and hence the controlled frequency response of SMEE plates. Moreover, the numerical results of this research lay a strong platform in the field of vibration and control. Also, it encourages the optimized design and analysis of smart SMEE structures for the application of sensors and actuators.

Elasto-thermoelectric beam formulation for modeling thermoelectric devices

The present paper provides a dynamic, non-linear and fully coupled Finite Element (FE) formulation based on the Timoshenko beam theory to study elasto-thermoelectric responses in thermoelectric devices. The two main motivations of this work are: i) to study mechanical responses in thermoelectric devices, which must be taken into account in the design of Peltier cells due to the fragility and relative low strength of the semiconductors, and ii) to provide a numerical tool that decreases the CPU time to allow the introduction of designs based on optimization processes and on sensitivity analyses that could require many evaluations. In order to undertake the objectives of this work, the general three-dimensional governing equations are reduced to one-dimensional ones by means of several assumptions. Then, a set of five multi-coupled partial differential equations is obtained. The resultant expressions are thermodynamically consistent and form a multi-coupled monolithic FE formulation, differently to stagger formulations that require two separated steps to reach the final result. Numerically, this set of multi-coupled equations is discretized using the FE method and implemented into FEAP Taylor, 2010 [1]. For a proper validation of the code, four benchmarks are performed using one-dimensional dynamic analytical solutions developed by the authors. Finally, this formulation is compared with a three-dimensional FE formulation also developed by the authors in Pe´rez-Aparicio et al., 2015 [2] to model a commercial Peltier cell. This comparison reveals that: i) relative errors are lower than 13% and ii) CPU times decrease significantly, more than one order of magnitude. In conclusion, the beam thermoelectric formulation is an accurate model that reduces CPU time and could be used in future design of thermoelectric devices.

3D modelling of heat transfer and moisture transport in young HPC structures with hybrid finite elements

Simulation of the hygro-thermo-chemical process of cement hydration in high-performance concrete (HPC) is performed coupling heat transfer with moisture transport, and accounting for an internal heat source and a moisture sink designed to simulate the chemical reactions developing in HPC structures. The three-dimensional finite element formulation is based on the direct approximation of temperature and relative humidity (RH) in the element domain. The heat and moisture flux fields are independently approximated on its boundary. Naturally hierarchical bases are used to set up these approximations to enhance adaptive refinement, while the geometry of the element is described independently to enhance the use of coarse, unstructured meshes. The solving system is sparse and well suited to parallelization. The relative performance of the formulation is illustrated using testing problems supported by experimental data.

Plasticity coupled with thermo-electric fields: Thermodynamics framework and finite element method computations

A consistent thermodynamic-based theoretical framework and three-dimensional finite element formulation is presented, capable of coupling elastic, thermal and electric fields. The complete set of governing equations is obtained from conservation principles for electric charge, energy and momentum. The second principle of thermodynamics is taken into account to introduce the irreversible phenomena, such as plastic dissipation or Joule heating. The constitutive relations are derived consistently from the Helmholtz free-energy potential for each corresponding dual variable in terms of the defined set of state variables. We consider the case of linear isotropic hardening model for plasticity, and provide the consistent form of the tangent thermo–electro–elastoplastic modulus through dual variable computations. The latter plays the crucial role in ensuring fast convergence properties of the finite element computations with the proposed coupled plasticity model. The implementation is carried out in a research version of the well-known computer code FEAP. Several numerical simulations are presented in order to illustrate the proposed model and formulation capabilities for providing an enhanced formulation of an important practical application in terms of Peltier cells.

A finite element framework for modeling internal frictional contact in three-dimensional fractured media using unstructured tetrahedral meshes

This paper introduces a three-dimensional finite element (FE) formulation to accurately model the linear elastic deformation of fractured media under compressive loading. The presented method applies the classic Augmented Lagrangian(AL)-Uzawa method, to evaluate the growth of multiple interacting and intersecting discrete fractures. The volume and surfaces are discretized by unstructured quadratic triangle-tetrahedral meshes; quarter-point triangles and tetrahedra are placed around fracture tips. Frictional contact between crack faces for high contact precisions is modeled using isoparametric integration point-to-integration point contact discretization, and a gap-based augmentation procedure. Contact forces are updated by interpolating tractions over elements that are adjacent to fracture tips, and have boundaries that are excluded from the contact region. Stress intensity factors are computed numerically using the methods of displacement correlation and disk-shaped domain integral. A novel square-root singular variation of the penalty parameter near the crack front is proposed to accurately model the contact tractions near the crack front. Tractions and compressive stress intensity factors are validated against analytical solutions. Numerical examples of cubes containing one, two, twenty four and seventy interacting and intersecting fractures are presented.

Validation of flexible multibody dynamics beam formulations using benchmark problems

As the need to model flexibility arose in multibody dynamics, the floating frame of reference formulation was developed, but this approach can yield inaccurate results when elastic displacements becomes large. While the use of three-dimensional finite element formulations overcomes this problem, the associated computational cost is overwhelming. Consequently, beam models, which are one-dimensional approximations of three-dimensional elasticity, have become the workhorse of many flexible multibody dynamics codes. Numerous beam formulations have been proposed, such as the geometrically exact beam formulation or the absolute nodal coordinate formulation, to name just two. New solution strategies have been investigated as well, including the intrinsic beam formulation or the DAE approach. This paper provides a systematic comparison of these various approaches, which will be assessed by comparing their predictions for four benchmark problems. The first problem is the Princeton beam experiment, a study of the static large displacement and rotation behavior of a simple cantilevered beam under a gravity tip load. The second problem, the four-bar mechanism, focuses on a flexible mechanism involving beams and revolute joints. The third problem investigates the behavior of a beam bent in its plane of greatest flexural rigidity, resulting in lateral buckling when a critical value of the transverse load is reached. The last problem investigates the dynamic stability of a rotating shaft. The predictions of eight independent codes are compared for these four benchmark problems and are found to be in close agreement with each other and with experimental measurements, when available.

Characterizing the mechanical properties of carbon nanocones using an accurate spring-mass model

A three-dimensional finite element (FE) formulation based on a spring-mass model is presented to investigate the mechanical properties of single-walled carbon nanocones (SWCNCs). The rotational spring elements together with longitudinal ones are employed for modeling the covalent bond between the carbon atoms, and the carbon atoms are modeled by mass elements. Analytical expressions for Young’s and shear moduli of SWCNCs with five feasible apex angles are obtained in terms of axial and torsion loads, respectively. The effects of geometrical parameters on the mechanical properties of SWCNCs are investigated. It is found that the apex angle of SWCNCs has a significant influence on their Young’s and shear moduli. Moreover, in contrast to the results of similar works in the literature, the present results reveal that the length and small radius of nanocones do not play a major role in their mechanical properties. It is shown that with increasing small radius, Young’s modulus slightly increases. To assess the accuracy of the developed FE formulation, molecular dynamics simulations are also conducted.

Three-dimensional finite element formulation for nonlinear dynamic analysis of seismic site and structure response

A three-dimensional finite element model for nonlinear dynamic analysis of seismic site and structure response is proposed and discussed. A series of numerical examples are presented which include modelling of a reinforced concrete frame with a portion of the ground consisting of various horizontal layers resting on rigid bedrock. The theoretical framework of the numerical analysis is based on continuum mechanics and irreversible thermodynamics. The spatial discretisation is performed by a combination of linear tetrahedral (ground) and hexahedral (structure) finite elements. The time integration is carried out by the leap-frog method. Total Lagrange formulation is adopted to account for large rotations and large displacements. To account for cracking and damage of the concrete, the frame structure is modelled by the microplane model. Damage and cracking phenomena are modelled within the concept of smeared cracks. Plasticity model is used for the modelling the reinforcement and the ground adopting the Von Mises and Drucker–Prager yield criteria, respectively. The influence of the ground layer configurations on the structure response is investigated and discussed. Comparative analysis shows the importance of the soil–structure interaction effects and material nonlinearity. Furthermore, with the implemented microplane model, which is aimed to be used for fracture and damage analysis of concrete, it is possible do assess the sustained structural damage.

Three-dimensional natural frequency analysis of anisotropic functionally graded annular sector plates resting on elastic foundations

Abstract Natural frequency analysis of anisotropic functionally graded material (FGM) annular sector plates on Winkler elastic foundations based on three-dimensional theory of elasticity was investigated. The three-dimensional graded finite element formulation was derived based on the principle of minimum potential energy and the Rayleigh-Ritz method. For an orthotropic FGM, the material properties were assumed to have in-plane polar orthotropy and transverse heterogeneity according to an exponential law, whereas the mass density was assumed to be constant. For an isotropic FGM, material properties varied continuously through the thickness direction according to a power-law distribution, whereas Poisson’s ratio was set to be constant. The effects of material gradient exponents, different sector angles, different thickness ratio, Winkler parameter and two different boundary conditions on the natural frequencies and mode shapes of FGM annular sector plates have been investigated. Numerical solution was compared with the result of an FGM annular circular plate, which showed good agreement.

An accurate spring-mass finite element model for vibration analysis of single-walled carbon nanotubes

An accurate spring-mass model, in the context of a three-dimensional finite element formulation, is developed for investigating the vibrational characteristics of single-walled carbon nanotubes (SWCNTs). Atoms are replaced by lumped mass elements at their locations and appropriate spring-type elements are defined as interconnections between the atoms in order to simulate the interatomic interactions. The effect of out of plane angle variation energy is incorporated into the model. The obtained results for the fundamental frequency of single-walled carbon nanotubes of various kinds are graphically illustrated. The influences of some commonly-used boundary conditions and changes in the nanotube geometrical parameters on vibration frequencies are examined. The numerical results show good agreement with other published results in the literature. Also, some novel relations are deduced which can be more useful in predicting the fundamental frequency of SWCNTs with great number of atoms.