Isogeometric Finite Element Research Articles

A nonlinear hyperelasticity model for single layer blue phosphorus based on ab initio calculations

A new hyperelastic membrane material model is proposed for single layer blue phosphorus ( β -P), also known as blue phosphorene. The model is fully nonlinear and captures the anisotropy of β -P at large strains. The material model is calibrated from density functional theory (DFT) calculations considering a set of elementary deformation states. Those are pure dilatation and uniaxial stretching along the armchair and zigzag directions. The DFT calculations are performed with the Quantum ESPRESSO package. The material model is compared and validated with additional DFT results and existing DFT results from the literature, and the comparison shows good agreement. The new material model can be directly used within computational shell formulations that are, for example, based on rotation-free isogeometric finite elements. This is demonstrated by simulations of the indentation and vibration of single layer blue phosphorus sheets at micrometer scales. The elasticity constants at small deformations are also reported.

Static analysis of piezoelectric functionally graded porous plates reinforced by graphene platelets

In this study, for the first time an isogeometric finite element formulation for bending analysis of functionally graded porous (FGP) plates reinforced by graphene platelets (GPLs) embedded in piezoelectric layers is presented. It is named as PFGP-GPLs for a short. The plates are constituted by a core layer, which contains the internal pores and GPLs dispersed in the metal matrix either uniformly or non-uniformly according to three different patterns, and two piezoelectric layers perfectly bonded on the top and bottom surfaces of host plate. The modified Halpin–Tsai micromechanical model is used to estimate the effective mechanical properties which vary continuously along thickness direction of the core layer. In addition, the electric potential is assumed to vary linearly through the thickness for each piezoelectric sublayer. A generalized C0-type higher-order shear deformation theory (C0-HSDT) in association with isogeometric analysis (IGA) is investigated. The effects of weight fractions and dispersion patterns of GPLs, the coefficient and distribution types of porosity as well as external electrical voltages on structure’s behaviors are investigated through several numerical examples.
 Keywords:
 piezoelectric materials; FG-porous plate; graphene platelet reinforcements; isogeometric analysis.

Fast and green parallel isogeometric analysis computations for multi-objective optimization of liquid fossil fuel reserve exploitation with minimal groundwater contamination

We present a general optimization technique to minimize the execution time, energy consumption, and numerical error of a parallel isogeometric finite element method (IGA-FEM) solver for time-dependent problems. The IGA-FEM solver is called upon multiple times during inverse problem computations. It is used to evaluate the inverse problem parameters. As an exemplary challenging problem, we consider the IGA-FEM solver as a primal solver for the inverse optimization of hydraulic fracking. Liquid Fossil Fuel Extraction Problem (LFFEP) is defined to maximize extraction and minimize contamination for the hydraulic fracking process. Solving the LFFEP problem requires hundreds of computationally demanding calls to the IGA-FEM solver. Thus, we research the optimal configuration of the IGA-FEM solver to make computations more efficient; i.e., to reduce computation time and energy consumption where the input parameters are the number of utilized CPUs in a parallel environment, mesh size, and B-spline order. The algorithm presented in this paper can be used to optimize any similar time-dependent IGA-FEM simulation.

Structural shape optimization of three dimensional acoustic problems with isogeometric boundary element methods

The boundary element method (BEM) is a powerful tool in computational acoustics, because the analysis is conducted only on structural surfaces, compared to the finite element method (FEM) which resorts to special techniques to truncate infinite domains. The isogeometric boundary element method (IGABEM) is a recent progress in the category of boundary element approaches, which is inspired by the concept of isogeometric analysis (IGA) and employs the spline functions of CAD as basis functions to discretize unknown physical fields. As a boundary representation approach, IGABEM is naturally compatible with CAD and thus can directly perform numerical analysis on CAD models, avoiding the cumbersome meshing procedure in conventional FEM/BEM and eliminating the difficulty of volume parameterization in isogeometric finite element methods. The advantage of tight integration of CAD and numerical analysis in IGABEM renders it particularly attractive in the application of structural shape optimization because (1) the geometry and the analysis can be interacted, (2) remeshing with shape morphing can be avoided, and (3) an optimized solution returns a CAD geometry directly without postprocessing steps. In the present paper, we apply the IGABEM to structural shape optimization of three dimensional exterior acoustic problems, fully exploiting the strength of IGABEM in addressing infinite domain problems and integrating CAD and numerical analysis. We employ the Burton–Miller formulation to overcome fictitious frequency problems, in which hyper-singular integrals are evaluated explicitly. The gradient-based optimizer is adopted and shape sensitivity analysis is conducted with implicit differentiation methods. The design variables are set to be the positions of control points which directly determine the shape of structures. Finally, numerical examples are provided to verify the algorithm.

Hybrid IG-FE method applied to cohesive fracture/contact in particle-filled elastomeric composites

In this paper, employing a new numerical framework, a 2D investigation is conducted on the effect of fiber-matrix contact/debonding on the stress–strain response of fiber-reinforced elastomeric composites. The analysis is carried out in the framework of large deformation/sliding. The main novelty of this work resides in using hybrid IsoGeometric-Finite Element (IG-FE) technique to discretize the domain where interfaces together with a band around them are represented by NURBS patches. These patches are coupled to the surrounding Lagrangian domain thanks to the transition elements. Interfaces’ discontinuity is readily induced through knot insertion capability within IGA. Moreover, contact constraints together with cohesive behavior are incorporated in the numerical model employing a unified mortar framework with appropriate definitions of potentials. Two numerical examples are considered within the proposed context and results are discussed in terms of stress–strain curves and stress contours. Due to proven higher accuracy of IGA in modeling contact/cohesive interfaces under large deformation and efficient use of NURBS around discontinuities, the proposed numerical methodology seems to be an appropriate (yet efficient) candidate to model such kind of highly nonlinear problems.

An isogeometric finite element formulation for phase transitions on deforming surfaces

This paper presents a general theory and isogeometric finite element implementation for studying mass conserving phase transitions on deforming surfaces. The mathematical problem is governed by two coupled fourth-order nonlinear partial differential equations (PDEs) that live on an evolving two-dimensional manifold. For the phase transitions, the PDE is the Cahn–Hilliard equation for curved surfaces, which can be derived from surface mass balance in the framework of irreversible thermodynamics. For the surface deformation, the PDE is the (vector-valued) Kirchhoff–Love thin shell equation. Both PDEs can be efficiently discretized using C1-continuous interpolations without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured spline spaces with pointwise C1-continuity are utilized for these interpolations. The resulting finite element formulation is discretized in time by the generalized-α scheme with adaptive time-stepping, and it is fully linearized within a monolithic Newton–Raphson approach. A curvilinear surface parameterization is used throughout the formulation to admit general surface shapes and deformations. The behavior of the coupled system is illustrated by several numerical examples exhibiting phase transitions on deforming spheres, tori and double-tori.

Preconditioning immersed isogeometric finite element methods with application to flow problems

Immersed finite element methods generally suffer from conditioning problems when cut elements intersect the physical domain only on a small fraction of their volume. We present a dedicated Additive-Schwarz preconditioner that targets the underlying mechanism causing the ill-conditioning of these methods. This preconditioner is applicable to problems that are not symmetric positive definite and to mixed problems. We provide a motivation for the construction of the Additive-Schwarz preconditioner, and present a detailed numerical investigation into the effectiveness of the preconditioner for a range of mesh sizes, isogeometric discretization orders, and partial differential equations, among which the Navier–Stokes equations.

Speeding up multi-objective optimization of liquid fossil fuel reserve exploitation with parallel hybrid memory integration

In this paper, we show a hybrid parallelization method for reducing the computational cost of a solver using hybrid-memory parallel machines. We show how the hybrid parallelization can be utilized to speed up challenging computational applications. Namely, we focus on the liquid fossil fuel reservoir exploitation problem (LFFEP), the optimization inverse problem where we try to find the optimal locations of pumps and sinks during the oil exploitation technique with hydraulic fracturing to maximize the amount of oil and minimize groundwater contamination. In our simulations, we combine a hierarchical genetic search (HGS) with an isogeometric finite element method solver.

Isogeometric thermal postbuckling of FG-GPLRC laminated plates

An analysis on thermal buckling and postbuckling of composite laminated plates reinforced with a low amount of graphene platelets is performed in the current investigation. It is assumed that graphaene platelets are randomly oriented and uniformly dispersed in each layer of the composite media. Elastic properties of the nanocomposite media are obtained by means of the modified Halpin-Tsai approach which takes into account the size effects of the graphene reinforcements. By means of the von Karman type of geometrical nonlinearity, third order shear deformation theory and nonuniform rational B-spline (NURBS) based isogeometric finite element method, the governing equations for the thermal postbuckling of nanocomposite plates in rectangular shape are established. These equations are solved by means of a direct displacement control strategy. Numerical examples are given to study the effects of boundary conditions, weight fraction of graphene platelets and distribution pattern of graphene platelets. It is shown that, with introduction of a small amount of graphene platelets into the matrix of the composite media, the critical buckling temperature of the plate may be enhanced and thermal postbuckling deflection may be alleviated.

Hydroelastic vibration analysis of plates partially submerged in fluid with an isogeometric FE-BE approach

This paper presents the hydroelastic vibration analysis of clamped rectangular plates, vertically or horizontally submerged in fluid by using isogeometric finite element and boundary element methods. The method of analysis is divided into two parts. In the first part, the dynamic characteristics of the structure, in vacuo conditions and in the absence of external forces, are obtained by NURBS-based isogeometric finite element method (IGAFEM). In the second part of the analysis, the fluid-structure interaction effects are calculated by using a NURBS-based isogeometric boundary element method (IGABEM) in conjunction with the method of images, in order to impose appropriate boundary condition on the fluid's free surface. By adopting the linear hydroelasticity theory, the fluid is assumed ideal, i.e., inviscid, incompressible and its motion is irrotational. The fluid-structure interaction effects are calculated in terms of the generalized added mass coefficients. In order to demonstrate the applicability of the proposed method, two different clamped rectangular plates were adopted for the calculations. The effects of the plate thickness and aspect ratio are also investigated. The predictions compare well with available numerical and experimental results found in the literature.

Computation of limit and shakedown loads for pressure vessel components using isogeometric analysis based on Lagrange extraction

This paper presents an isogeometric finite element method based on Lagrange extraction in combination with a primal-dual algorithm for estimating limit and shakedown loads of pressure vessel components. The primal-dual algorithm is used to determinate simultaneously both upper and quasi-lower bounds of the limit load multiplier. The Lagrange extraction is integrated into isogeometric analysis. This way, which directly connects a C0 Lagrange basis with smooth NURBS basis and offers an alternative implementation of isogeometric analysis in the standard finite element method code, allows computations in the same manner as the standard finite element, but the present approach possesses the distinguishing advantages of using higher-order basis approximations. Numerical results of plane and axisymmetric problems are compared with analytical or other available solutions to prove the reliability and efficiency of the present approach. The application of shakedown analysis in pressure vessel engineering is illustrated by some examples.

Porosity-dependent nonlinear transient responses of functionally graded nanoplates using isogeometric analysis

The study using numerical methods on porous functionally graded (FG) nanoplates is still somewhat limited. This paper focuses on porosity-dependent nonlinear transient analysis of FG nanoplates using isogeometric finite element approach. In order to capture the small size effects, the Eringen's nonlocal elasticity based on higher order shear deformation theory (HSDT) are used to model the porous FG nanoplates. Two distributions of porosities inside FG materials are incorporated and defined via a modified rule of mixture. The nonlinear transient nonlocal governing equations under transverse dynamic loads are formulated by using the von Kármán strains and are solved by Newmark time integration scheme to obtain geometrically nonlinear responses. It is indicated that nonlinear transient deflections of the porous FG nanoplate are significantly influenced by material composition, porosity, nonlocal parameters, volume fraction exponent, porosity distributions, geometrical parameters and dynamic load characteristics.

Dynamics of a capsule flowing in a tube under pulsatile flow

We analyze numerically the behavior of a deformable micro-capsule confined in a pipe under a pulsatile flow. The capsule moves and is deformed by the action of a pulsatile flow inside the tube with a non-null mean velocity. This configuration can be found in the nature and in many bioengineering systems where artificial capsules are driven by micro-pumps through micro-channels. The capsule is considered as a thin hyperelastic membrane, which encloses an internal fluid. As it has been demonstrated in the literature, this model represents a wide range of artificial capsules, for example, the alginate-based capsules, typically used in bioengineering applications. A hybrid isogeometric finite element method and boundary element method based on a T-spline discretization and formulated in the time domain is used to solve the mechanical and hydrodynamical equations. The influence of the relative rigidity of the membrane, frequency and amplitude of the pulsatile flow is studied. Results show that the behavior of the capsule differs from steady flows and it depends strongly on the frequency of the flow and mechanical characteristic of the capsule.

Hybrid Isogeometric-Finite Element Discretization Applied to Stress Concentration Problems

Isogeometric analysis (IGA) employs non-uniform rational B-splines (NURBS) or other B-spline-based variants to represent both the geometry and the field variable. Exact geometry representation and higher order global continuity (at least [Formula: see text] even on elements’ boundaries) are two favorable properties that would make IGA an appropriate discretization technique in problems with responses associated with the derivatives of the primary field variable. As a category of these problems, in this paper, 2D elastostatic problems involving stress concentration sites are analyzed with a hybrid isogeometric-finite element (IG-FE) discretization. To exploit higher order continuity of NURBS basis functions, IGA discretization is applied selectively at pre-identified locations of high displacement gradients where the stress concentration occurs. In addition, considering computational efficiency, the rest of problem domain is discretized by means of linear Lagrangian finite elements. The connection of NURBS and Lagrangian domain is carried out through employing specially devised elements [Corbett, C. J. and Sauer, R. A. [2014] “NURBS-enriched contact finite elements”, Computer Methods in Applied Mechanics and Engineering 275, 55–75]. The methodology is applied in some 2D elastostatic examples. Increasing the number of DOFs and comparing convergence of the concentrated stress value using different discretizations, it is shown that the hybrid IG-FE discretizations generally have faster and more stable convergence response compared with pure FE discretizations especially at lower DOFs.

Parallel fast isogeometric L2 projection solver with GALOIS system for 3D tumor growth simulations

We present a parallel solver for isogeometric L2 projections implemented in GALOIS system. We use the solver for isogeometric finite element method simulations of tumor growth. First, we describe the dependencies between the quantities describing the melanoma tumor growth in three-dimensions. Later, we derive the system of partial differential equations modeling the tumor dynamics, including tumor cell density, tumor angiogenic factor, flux, pressure, extracellular and degraded extracellular matrices. The continuous model is coupled every ten-time steps with graph grammar model describing the vasculature network. The resulting system of PDEs is solved with our parallel solver over the shared memory Linux cluster node. We show high computational efficiency and accuracy of our model and we demonstrate its perfect parallel scalability with the number of CPU cores. We discuss the application of our solver in predictive oncology.

Supersonic panel flutter of variable stiffness composite laminated skew panels subjected to yawed flow by using NURBS-based isogeometric approach

An enhanced isogeometric finite element method is developed to investigate the free vibration analysis as well as the linear flutter characteristics of finite square as well as skew laminated plates. The laminates are assumed composed of variable stiffness plies due to implementation of curvilinear fibers. The flutter analysis of the skew laminate under various flow yaw angles are also taken into account. The first-order shear deformation plate structural assumptions are utilized in conjunction with an aerodynamic loading model according to the first-order piston theory. The cubic NURBS basis functions are employed to build the plate’s geometry while are also hired as the shape functions for finite element analysis field approximations. In order to show the accuracy and effectiveness of the present formulation, some representative results are provided and compared to those available in the literature. The flutter characteristics of curvilinear skew laminates are investigated and the effects of different changing parameters including boundary condition, skewness, flow direction and layup on the dynamic stability of the panel from both free vibration as well as flutter points of view are then studied.

A monolithic fluid–structure interaction formulation for solid and liquid membranes including free-surface contact

A unified fluid–structure interaction (FSI) formulation is presented for solid, liquid and mixed membranes. Nonlinear finite elements (FE) and the generalized-α scheme are used for the spatial and temporal discretization. The membrane discretization is based on curvilinear surface elements that can describe large deformations and rotations, and also provide a straightforward description for contact. The fluid is described by the incompressible Navier–Stokes equations, and its discretization is based on stabilized Petrov–Galerkin FE. The coupling between fluid and structure uses a conforming sharp interface discretization, and the resulting non-linear FE equations are solved monolithically within the Newton–Raphson scheme. An arbitrary Lagrangian–Eulerian formulation is used for the fluid in order to account for the mesh motion around the structure. The formulation is very general and admits diverse applications that include contact at free surfaces. This is demonstrated by two analytical and three numerical examples exhibiting strong coupling between fluid and structure. The examples include balloon inflation, droplet rolling and flapping flags. They span a Reynolds-number range from 0.001 to 2000. One of the examples considers the extension to rotation-free shells using isogeometric FE.

Form II of Mindlin's second strain gradient theory of elasticity with a simplification: For materials and structures from nano- to macro-scales

Abstract The fundamental equations for Form II of Mindlin's second strain gradient elasticity theory for isotropic materials are first derived. A corresponding simplified formulation is then proposed, with six and two higher-order material parameters for the strain and kinetic energy, respectively. This simplified model is still capable of accounting for free surface effects and surface tension arising in second strain gradient continua. Within the simplified model, at first, surface tension effects appearing in nano-scale solids near free boundaries are analyzed. Next, a thin strip under tension and shear is considered and closed-form solutions are provided for analyzing the free surface effects. Expressions for effective Poisson's ratio and effective shear modulus are proposed and found to be size-dependent. Most importantly, for each model problem a stability analysis is accomplished disallowing non-physical solutions (befallen but not exclusively disputed in a recent Form I article). Finally, a triangular macro-scale lattice structure of trusses is shown, as a mechanical metamaterial, to behave as a second strain gradient continuum. In particular, it is shown that initial stresses prescribed on boundaries can be associated to one of the higher-order material parameters, modulus of cohesion, giving rise to surface tension. For completeness, a numerical free vibration eigenvalue analysis is accomplished for both a fine-scale lattice model and the corresponding second-order continuum via standard and isogeometric finite element simulations , respectively, completing the calibration procedure for the higher-order material parameters. The eigenvalue analysis confirms the necessity of the second velocity gradient terms in the kinetic energy density .

Bending and free vibration analyses of in-plane bi-directional functionally graded plates with variable thickness using isogeometric analysis

This article is firstly concerned with bending and free vibration analyses of in-plane bi-directional functionally graded (IBFG) plates with variable thickness in the framework of isogeometric analysis (IGA). The plate thickness is smoothly altered in both x- and y-axes by a predetermined power law. Two types of power-law material models with the symmetrical and asymmetrical volume fraction distribution are suggested to characterize the in-plane material inhomogeneity. A non-uniform rational B-spline (NURBS) surface for simultaneously representing both variable thickness and volume fraction distribution of each constituent is employed. By using the k-refinement strategy, the C0-continuous requirement at symmetrical material interfaces can be achieved, yet still ensuring material gradations elsewhere owing to the prominent advantage of NURBS basis functions in easily controlling continuity. Effective material properties are then evaluated by either the rule of mixture or the Mori-Tanaka scheme. An analysis NURBS surface separately created with the foregoing NURBS surface is utilized to exactly describe geometry and approximately solve unknown solutions in finite element analysis (FEA) based on the IGA associated with a generalized shear deformation theory (GSDT). The Galerkin C1-continuous isogeometric finite element model is therefore simply achieved due to the possibility of flexibly meeting high-order derivatives and continuity of analysis NURBS functions. In addition, no shear correction factors exist in the present formulation, although shear deformation effects are still considered. The influences of variable thickness, material property, length-to-thickness ratio, boundary condition on bending and free vibration responses are investigated and discussed in detail through several numerical examples.

NURBS-based isogeometric thermal postbuckling analysis of temperature dependent graphene reinforced composite laminated plates

A thermal postbuckling analysis for composite laminated plates reinforced with graphene sheets is performed in this research. All of the thermomechanical properties of the composite media are assumed to be temperature dependent. Volume fraction of the graphene in each layer is assumed to be different which results in a piecewise functionally graded plate. Based on the third order shear deformation plate theory of Reddy, the total strain energy of the plate is obtained. Composite laminated plate is assumed to be under uniform temperature rise. Properties of the graphene reinforced composite media are estimated by means of a refined Haptin-Tsai approach which contains efficiency parameters to capture the size dependency of the constituents. Afterwards, a non-uniform rational B-spline (NURBS) based isogeometric finite element method is implemented to study the thermal postbuckling response of the graphene reinforced composite laminated plates. Thermally induced postbuckling curves of the composite plate reinforced by graphene are provided for different functionally graded patterns, aspect ratios, side to thickness ratios and boundary conditions. It is shown that, FG-X pattern of graphene reinforcement results in the highest critical buckling temperature and the lowest postbuckling deflection.