Unknown Field Variables Research Articles

A dynamic description of material brittle failure using a hybrid phase-field model enhanced by adaptive isogeometric analysis

The objective of this work is to study dynamic crack propagation in brittle materials under time-dependent loading conditions by using the recently developed adaptive isogeometric phase-field approach. The current approach owns several ingredients including the advantages of the phase-field method (PFM), which can be used to model complex crack morphologies without any fracture criterion, and the isogeometric analysis, which possesses some excellent features such as exact geometry and high-order continuity. In addition, the model is further enhanced by employing the locally refined non-uniform rational B-spline (LR NURBS) basis, which is utilized for spatial discretization and phase-field discretization, while the generalized-α or HHT-α method is employed for the temporal discretization. For the coupled elasto-phase field system, a staggered approach is adopted to compute the unknown field variables. Moreover, a hybrid formulation of the PFM is employed to recover the linearity of momentum equation. The structured mesh adaptive refinement is conducted based on the prescribed threshold on the phase-field variable, thus it can effectively alleviate the burden of computational overhead, which is the main shortcoming of the PFM. Three problems involving dynamic crack growth is solved to show the robustness and the applicability of the proposed approach. It is shown that the proposed approach can obtain accurate results at a fraction of degrees-of-freedom without compromising accuracy in comparison with uniformly refined mesh.

Numerical study on thermal enhancement in magnetohydrodynamic micropolar liquid subjected to motile gyrotactic microorganisms movement and Soret and dufour effects

Motile gyrotactic organism activity in micropolar fluid subjected to simultaneous heat and mass transfer in the presence of thermal diffusion and diffusion thermo effects is modeled. The base fluid is taken as kerosene oil. The immersion of nano-sized particles of C u − T i O 2 kerosene oil makes its rheological characteristics of micropolar fluid because nano-sized particles serve as micro-structures. In this case couple stress, vortex and spin gradient viscosities become significant and simultaneously linear and angular momenta become equally remarkable. So, micropolar theory is the best to model the immersion of nano-structures in the fluid. Similarity variables are used to transform the problems into their dimensionless forms. The mathematical models are solved numerically by applying the finite element method (FEM). The results are validated and numerical experiments are performed to analyze the key parameters of the unknown field variables. In order to analyze the impact of physical parameters on field variables, numerous simulations are performed. The micro-rotation of microparticles and nanoparticles due to fluid deformation motivates fluids particles to rotate faster. Therefore, linear motion is observed to be increased for higher values of vortex viscosity. The curvature parameter has significantly impacted the linear motion of the fluid particles. The impact of the curvature parameter on the movement of nanofluid is lesser than the impact of the curvature parameter on the motion of hybrid nanoparticles. Concentration gradient supports the diffusion of heat in the fluid regime and numerical experiments with variation of Dufour (that measures the effects of concentration gradient on heat diffusion) have shown that concentration gradient is more effective in the case of hybrid nanofluid relative to the mono nanofluid. The angular motion of fluid particles increases when vortex viscosity is increased. Numerical simulations have predicted that the angular motion of particles of hybrid nanofluid has a lesser angular speed than that of the angular motion of particles of mono nanofluid. The Soret number determines the impact of temperature gradient on the diffusion of solute in the fluid. Simulations for various values of Soret number have predicted that the impact of temperature gradient on the diffusion of solute in hybrid nanofluid is stronger than the impact of temperature gradient on the diffusion of solute in mono nanofluid.

Toward identifying the ASR-induced stresses from displacement measurements and crack observations—Demonstration on a beam bridge in Norway

Alkali–silica reaction (ASR) in concrete causes expansion and degradation of the material, which might give adverse structural consequences. From the structural engineer point view, the greatest concern is if ASR leads to loss of structural integrity. Two natural questions arise when assessing existing concrete structures affected by ASR: (1) how to calculate the ASR-induced stresses, and (2), when the ASR-induced stresses are calculated, what is the residual capacity when accounting for the material deterioration caused by ASR? This study aimed to contribute in answering the first question.The ASR-induced stresses can be calculated in a structural analysis that includes a concrete material model that incorporates the effects of ASR on the material behaviour, i.e. expansion and material deterioration. Many such models exist. However, these models rely on predefined field variables, e.g. moisture and temperature, which are (generally speaking) unknowns for an existing structure. Consequently, structural analysis of ASR-affected concrete structures involves dealing with unknown field variables. From this background, we developed a material model and a suitable structural analysis method. The material model relies on only one predefined, howbeit unknown, field variable —the free ASR expansion. The structural analysis method is based on solving an inverse problem, which is to back-calculate the free ASR expansion field from a set of measured displacements.The material model and the structural analysis method were applied in a structural analysis of an ordinary reinforced beam bridge in Norway. Then, the imposed deformations and stresses due to ASR were investigated to increase the understanding of the structural consequences of ASR in ordinary reinforced continuous beam bridges.

Computational analysis of the nonlinear vibrational behavior of perforated plates with initial imperfection using NURBS-based isogeometric approach

Abstract In this article, an isogeometric analysis through NURBS basis functions is presented to study the nonlinear vibrational behavior of perforated plates with initial imperfection. In this regard, the governing equations of plate dynamics, as well as the displacement–strain relations, are derived using the Mindlin–Reissner plate theory by considering von Karman nonlinearity. The geometry of the structure is formed by selecting the order of NURBS basis functions and the number of control points according to the physics of the problem. Since similar basis functions are utilized to estimate the accurate geometry and displacement field of the domain, the order of the basic functions and the number of control points are optimized for the proper approximation of the unknown field variables. By utilizing the energy approach and Hamilton principle and discretizing the equations of motion, the vibrational response of the perforated imperfect plate is extracted through an eigenvalue problem. The results of linear vibrations, geometrically nonlinear vibrations, and nonlinear vibrations of imperfect plates are separately validated by considering the previously reported findings, which shows a satisfactory agreement. Thereafter, a coefficient of the first mode shape is considered as the initial imperfection and the vibrational analysis is reexamined. Furthermore, the nonlinear vibrations of the perforated plate with initial imperfection are analysed using an iterative approach. The effects of the perforated hole, initial imperfection, and geometric nonlinearity are also addressed and discussed.

Physics-based reduced order model for computational geomechanics

Numerical models are an essential tool in stability analysis, design, and construction for geotechnical engineering. Yet, numerical modeling is too time-consuming in practical engineering. In this study, a physics-based reduced order model (ROM) was developed to approximate the displacement and stress field of geotechnical structure by combining Latin hypercube sampling (LHS), a numerical method and proper orthogonal decomposition (POD). The set of design variables were constructed using LHS. A numerical model was used to generate the snapshots based on the design variables. POD was used to compute the eigenvalues and eigenvectors of a spatial Gram matrix, which was constructed based on snapshots. The first K eigenfunction vectors were determined based on eigenvalues and eigenvectors of the spatial Gram matrix. The interpolation matrix of elements was computed using a radial basin function (RBF), and then the vector of an element was determined by solving the penalized linear systems. To determine the new design variables, the coefficient of ROM was determined based on the RBF and the vector of elements, and the unknown field variables were predicted based on the ROM. The ROM was illustrated and verified for a circular tunnel. The results showed that the predicted displacement and stress field were in excellent agreement with both the analytical and numerical solutions. The physics-based ROM characterized well the deformation and failure mechanism of the surrounding rock mass and can be used to replace a numerical model for back analysis, optimal design, and uncertainty analysis of geotechnical engineering, thereby eliminating costly repetitive computations.

Mixed meshless local Petrov–Galerkin (MLPG) collocation methods for gradient elasticity theories of Helmholtz type

Two mixed meshless collocation methods for solving problems by considering a linear gradient elasticity theory of the Helmholtz type are proposed. The solution process is facilitated by employing operator-split procedures, splitting the original 4th-order problem into two uncoupled second-order sub-problems, which are then solved in a staggered manner by applying the mixed meshless local Petrov–Galerkin collocation strategy. Thereby, identical nodal pattern is used for the discretization of both lower-order problems, while the approximation of all unknown field variables is performed by applying the Moving Least Squares functions with interpolatory conditions. The performance of the derived methods is tested by some suitable numerical examples, dealing with elasticity problems that are often treated by gradient theories. Therein, it is demonstrated that both proposed methods are able to capture the size effect and that the strain-based method is able to produce the non-singular strain field around the crack tip, similar to the nonlocal elasticity approach of Eringen. It has been found out that the obtained results agree well with available analytical and numerical solutions.

On the application of polygonal finite element method for Stokes flow – A comparison between equal order and different order approximation

In this work, we discuss the application of polygonal finite elements to the solution of Stokes equations in two dimensions. The proposed framework is based on the mixed finite element discretization that involves velocity and pressure as independent unknown field variables. We present two approaches: (a) same order of approximation for velocity and pressure with Pressure Stabilization Petrov-Galerkin method (PSPG) and (b) different order of approximation for velocity and pressure. The approximation functions over polygons are rational polynomials based on Wachspress coordinates and for numerical integration of the terms in the bilinear and the linear form, we employ the linear smoothing technique. The relative performance between the approaches, the convergence properties and the accuracy is presented for two dimensional numerical examples, which shows that the proposed framework yields accurate results and converges at optimal convergence rate.

A four-variable global–local shear deformation theory for the analysis of deep curved laminated composite beams

A precise global–local shear deformation theory is developed for the prediction of static and dynamic behaviors of thin and thick layered curved beams. The effect of deepness is considered in the derivation of the proposed beam theory. Variations of the shear stress along the thickness direction of the curved beam are approximated by using a global parabolic shear stress function which is locally refined at each layer. The zero conditions of shear stresses on the boundary surfaces of the curved beam are exactly satisfied, and no shear correction coefficient is needed. One of the important features of the present theory is that it has only four unknown field variables, which is only one more than the first-order shear deformation theory. A displacement-based finite element model is employed for solving the governing equations. For validation, the results obtained from static and free vibration tests are compared with the results of three-dimensional (3D) finite element analysis, classical theories, and other advanced shear deformation beam theories. The obtained numerical results show that the present model can precisely predict static and free vibration responses of both shallow and deep composite beams with arbitrary boundary and layup conditions.

Adaptive phase field method for quasi-static brittle fracture using a recovery based error indicator and quadtree decomposition

An adaptive phase field method is proposed for crack propagation in brittle materials under quasi-static loading. The adaptive refinement is based on the recovery type error indicator, which is combined with the quadtree decomposition. Such a decomposition leads to elements with hanging nodes. Thanks to the polygonal finite element method, the elements with hanging nodes are treated as polygonal elements and hence do not require any special treatment. The mean value coordinates are used to approximate the unknown field variables and a staggered solution scheme is adopted to compute the displacement and the evolution of the phase field variable. A few standard benchmark problems are solved to show the efficiency of the proposed framework. It is seen that the proposed framework yields comparable results at a fraction of the computational cost when compared to standard approaches reported in the literature.

Meshless local B-spline collocation method for heterogeneous heat conduction problems

Several numerical issues still pertain in the modeling of heterogeneous heat conduction, particularly from the viewpoints of material discontinuity and its handling and the presence of heat source. In this paper, a meshless local B-spline collocation method is presented for unsteady heat conduction problems of heterogeneous media. Unknown field variables are approximated by using B-spline basis functions within overlapped compact domains covering the geometry of materials. The present method is a truly meshless approach. The proposed approach is mainly coming with the following advantage that it is straightforward in dealing with discontinuity across the interface of heterogeneous materials. Treatment of discontinuity by using non-crossing interface compact domains allows material discontinuity to be handled geometrically without enforcing additional term/function at the interface. Several heat conduction problems in 2D and 3D heterogeneous media with arbitrary discontinuity shapes are considered. Attention is given for heterogeneous heat conduction problem accompanied by the presence of crack as well. The analysis is then completed by simulating effect of heat generation, in particular which produces high temperature rise inside a heterogeneous structure/component. Simulation results show that the proposed method is a simple and accurate numerical technique for solving unsteady heat conduction problems of heterogeneous media.

A scaled boundary finite element formulation for poroelasticity

SummaryThis paper develops the scaled boundary finite element formulation for applications in coupled field problems, in particular, to poroelasticity. The salient feature of this formulation is that it can be applied over arbitrary polygons and/or quadtree decomposition, which is widely employed to traverse between small and large scales. Moreover, the formulation can treat singularities of any order. Within this framework, 2 sets of semianalytical, scaled boundary shape functions are used to interpolate the displacement and the pore fluid pressure. These shape functions are obtained from the solution of vector and scalar Laplacian, respectively, which are then used to discretise the unknown field variables similar to that of the finite element method. The resulting system of equations are similar in form as that obtained using standard procedures such as the finite element method and, hence, solved using the standard procedures. The formulation is validated using several numerical benchmarks to demonstrate its accuracy and convergence properties.

Review of emissions from smouldering peat fires and their contribution to regional haze episodes

Smouldering peat fires, the largest fires on Earth in terms of fuel consumption, are reported in six continents and are responsible for regional haze episodes. Haze is the large-scale accumulation of smoke at low altitudes in the atmosphere. It decreases air quality, disrupts transportation and causes health emergencies. Research on peat emissions and haze is modest at best and many key aspects remain poorly understood. Here, we compile an up-to-date inter-study of peat fire emission factors (EFs) found in the literature both from laboratory and from field studies. Tropical peat fires yield larger EFs for the prominent organic compounds than boreal and temperate peat fires, possibly due to the higher fuel carbon content (56.0 vs 44.2%). In contrast, tropical peat fires present slightly lower EFs for particulate matter with diameter ≤2.5 μm (PM2.5) for unknown reasons but are probably related to combustion dynamics. An analysis of the modified combustion efficiency, a parameter widely used for determining the combustion regime of wildfires, shows it is partially misunderstood and highly sensitive to unknown field variables. This is the first review of the literature on smouldering peat emissions. Our integration of the existing literature allows the identification of existing gaps in knowledge and is expected to accelerate progress towards mitigation strategies.

An efficient partial mixed finite element model for static and free vibration analyses of FGM plates rested on two-parameter elastic foundations

In this study, a four-node quadrilateral partial mixed plate element with low degrees of freedom (dofs) is developed for static and free vibration analysis of functionally graded material (FGM) plates rested on Winkler-Pasternak elastic foundations. The formulation of the presented finite element model is based on a parametrized mixed variational principle which is developed recently by the first author. The presented finite element model considers the effects of shear deformations and normal flexibility of the FGM plates without using any shear correction factor. It also fulfills the boundary conditions of the transverse shear and normal stresses on the top and bottom surfaces of the plate. Beside these capabilities, the number of unknown field variables of the plate is only six. The presented partial mixed finite element model has been validated through comparison with the results of the three-dimensional (3D) theory of elasticity and the results obtained from the classical and high-order plate theories available in the open literature.

Simulation of dynamic and static thermoelastic fracture problems by extended nodal gradient finite elements

In this paper, the recent development of extended consecutive-interpolation 4-node quadrilateral element (XCQ4), which goes beyond certain limitations of conventional methods, is further employed to study dynamic and static thermoelastic fracture problems. In particular, static stress intensity factors (SIFs) and transient dynamic responses of two-dimensional (2-D) stationary cracks in elastic solids under thermal and thermal-mechanical loadings are investigated. In addition, simulation of quasi-static crack propagation under thermal-mechanical loading condition is also performed. To precisely capture the singularity and discontinuity of temperature, heat flux, and displacements due to the presence of crack, the unknown field variables of mechanical displacements and temperature are approximated in terms of the XCQ4 framework by different enrichment strategies, which include either the use of the conventional branch functions or the new ramp function associated with Heaviside function. Only adiabatic condition is considered on the crack surfaces. For quasi-static crack growth analysis, the maximum hoop stress criterion, which is also valid for thermo-elastic fracture mechanics, is adopted. For transient crack analyses, the inertial effect is taken into account for the interaction integral in the consideration of DSIFs. Through eight numerical examples for thermoelastic fracture problems, the validity of the XCQ4 is confirmed by a comparison of the obtained results to those derived from different approaches in previous studies. In addition, the accuracy, performance, and superior advantages of the XCQ4 over standard extended finite element approach are also demonstrated.

A combination of isogeometric technique and scaled boundary method for the solution of the steady-state heat transfer problems in arbitrary plane domain with Robin boundary

The scaled boundary finite element method (SBFEM) has attracted considerable attention in recent years as a novel semi-analytical computational approach since only the boundary is discretized using finite element approach and the spatial dimension is reduced by one in this method. By introducing the radial and circumferential scaled boundary coordinates in this method, the reduction of spatial dimension is accomplished by using the conventional Lagrange functions to weaken the governing equations in the circumferential direction, while to work analytically in the radial direction. The NURBS-based isogeometric analysis (IGA) has remarkable advantages due to its integration of CAD/CAE, geometrical exact discretization for free-form shapes and numerical accuracy. Thus, an isogeometric analysis based on the framework of scaled boundary method (IGA-SBM) is proposed, which combines the concepts of IGA and SBFEM by employing NURBS to represent the unknown field variables in the circumferential direction. In this work, the IGA-SBM is extended to the solutions of the steady-state heat transfer problems in arbitrary plane domain enclosed by the complex boundary geometry. Four numerical examples are presented to illustrate that the excellent accuracy, computational efficiency and convergence performance of IGA-SBM. The numerical studies show that the heat transfer problems with complicated configuration can be more effectively handled by considering the combination of IGA and SBFEM.

Bending analysis of laminated composite plates using isogeometric collocation method

Isogeometric collocation has emerged as an efficient numerical technique for solving boundary value problems and is a potential alternative for Galerkin based Isogeometric methods. In this investigation, Isogeometric collocation has been proposed for the linear static bending analysis of laminated composite plates governed by Reissner–Mindlin theory. Three formulations are presented in this paper namely, standard primal formulation, mixed formulation and a locking-free primal formulation. The standard primal formulation adopts displacements and rotations as unknown field variables. Mixed formulation considers displacements, rotations and transverse shear forces as the unknown field variables. Locking-free primal formulation is a rotation-free formulation with displacements and transverse shear strains as the unknown field variables. Results for benchmark problems on bending of rectangular laminated composite plates are obtained and compared with the ones existing in the literature. The three formulations of Isogeometric collocation presented in this paper are assessed in terms of accuracy and the computational time required to assemble and solve the problem.

Mixed meshless local Petrov–Galerkin collocation method for modeling of material discontinuity

A mixed meshless local Petrov---Galerkin (MLPG) collocation method is proposed for solving the two-dimensional boundary value problem of heterogeneous structures. The heterogeneous structures are defined by partitioning the total material domain into subdomains with different linear-elastic isotropic properties which define homogeneous materials. The discretization and approximation of unknown field variables is done for each homogeneous material independently, therein the interface of the homogeneous materials is discretized with overlapping nodes. For the approximation, the moving least square method with the imposed interpolation condition is utilized. The solution for the entire heterogeneous structure is obtained by enforcing displacement continuity and traction reciprocity conditions at the nodes representing the interface boundary. The accuracy and numerical efficiency of the proposed mixed MLPG collocation method is demonstrated by numerical examples. The obtained results are compared with a standard fully displacement (primal) meshless approach as well as with available analytical and numerical solutions. Excellent agreement of the solutions is exhibited and a more robust, superior and stable modeling of material discontinuity is achieved using the mixed method.

Scattering of elastic P- and SV-waves by a circular coated nanofiber based on Gurtin–Murdoch model of surface elasticity: Scattering cross section results

ABSTRACTIn this article, an analytical study of elastic P- and SV-wave scattering by a circular nanofiber is presented. The nanofiber is assumed to be surrounded by an inhomogeneous interphase layer, and Gurtin–Murdoch's model of surface elasticity is utilized to study the surface/interface effects in the regions between the fiber and interphase and also interphase and matrix. The simultaneous effects of surface elasticity and interphase inhomogeneity are considered here; by taking the inhomogeneous interphase to be composed of several sublayers, a transfer matrix approach is used to find the unknown field variables and, consequently, the scattering cross sections. The results indicate that considering the effects of surface elasticity and interphase inhomogeneity has a considerable impact on the calculated scattering cross sections.

Isogeometric Analysis of Laminated Composite Plates Using the Higher-Order Shear Deformation Theory

Isogeometric analysis (IGA) aims at simplifying the computer aided design (CAD) and computer aided engineering (CAE) pipeline by using the same functions to describe the geometry (CAD) and the unknown fields (Analysis). IGA can be based on a variety of CAD descriptions, the most widely used today being non-uniform rational B-splines (NURBS). In this article, the suitability of NURBS-based isogeometric analysis within a third-order shear deformation theory for the simulation of the static, dynamic, and buckling response of laminated composite plates is investigated. The method employs NURBS basis functions to both represent the geometry (exactly) and the unknown field variables. One of the main advantages of the present method is directly inherited from IGA, that is to easily increase the approximation order. To avoid using a shear correction factor, a third-order shear deformation theory (TSDT) is introduced. It requires C1-continuity of generalized displacements and the NURBS basis functions are well suited for this requirement. Several numerical examples are used to demonstrate the performance of the present method compared with other published ones.

Torsional flexural steady state response of monosymmetric thin-walled beams under harmonic loads

Starting with Hamilton's variational principle, the governing field equations for the steady state response of thin-walled beams under harmonic forces are derived. The formulation captures shear deformation effects due to bending and warping, translational and rotary inertia effects and as well as torsional flexural coupling effects due to the cross section mono-symmetry. The equations of motion consist of four coupled differential equations in the unknown displacement field variables. A general closed form solution is then developed for the coupled system of equations. The solution is subsequently used to develop a family of shape functions which exactly satisfy the homogeneous form of the governing field equations. A super-convergent finite element is then formulated based on the exact shape functions. Key features of the element developed include its ability to (a) isolate the steady state response component of the response to make the solution amenable to fatigue design, (b) capture coupling effects arising as a result of section mono-symmetry, (c) eliminate spatial discretization arising in commonly used finite elements, (d) avoiding shear locking phenomena, and (e) eliminate the need for time discretization. The results based on the present solution are found to be in excellent agreement with those based on finite element solutions at a small fraction of the computational and modelling cost involved.