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Computational Mechanics (14320924)
Peridynamics (PD) stands out as a promising approach for addressing problems featuring cracks or strong discontinuities, surpassing methods grounded in classical continuum mechanics (CCM) within such contexts. Nonetheless, PD exhibits a heightened computational demand compared to CCM-based approaches, attributed to the nonlocal nature of the PD approaches and the application of a single-point integration rule in its standard discretization. This paper introduces an improved class of numerical quadratures to mitigate the latter challenge. Additionally, a reduced form of these numerical quadratures is proposed to diminish computational costs. A systematic approach for comparing different reduced patterns is presented. To extend the applicability of these numerical quadratures to regions near boundaries or cracks, an extended weighted least squares (WLS)-based numerical integration scheme is introduced. The performance of the proposed schemes is investigated through a series of numerical examples. The findings indicate that the proposed approach demonstrates strong performance across various challenges, such as crack propagation in 3D domains and multiphysical scenarios. © The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2025.
Mossaiby, F.,
Häfner, G.,
Shojaei a., A.,
Hermann, A.,
Cyron, C.,
Müller, M.,
Silling, S. Computer Methods in Applied Mechanics and Engineering (00457825)439
This study introduces a computational framework for simulating the self-assembly of diblock copolymers using a novel peridynamic (PD)-enhanced Fourier spectral method (FSM). Diblock copolymers, composed of two distinct polymer blocks, are capable of forming nanostructured domains with applications in nanoelectronics, photonics, and advanced membranes. Current simulation techniques face challenges in capturing the multiscale dynamics of polymer systems and are often limited by computational inefficiencies. Our approach combines a phase-field model with FSM for spatial discretization and leverages a PD-based diffusion operator to overcome the stability restrictions of explicit time-stepping schemes. This integration allows for larger time steps, ensuring both stability and computational efficiency. The method's scalability is enhanced through parallel implementation using C++ and OpenMP, optimized for multi-core CPUs. Validation through phase diagrams of copolymer melts and simulations of evaporation-induced self-assembly (EISA) processes demonstrates the capability of the proposed method to accurately capture large-scale, dynamic morphologies. Our approach provides a versatile framework and was found in certain examples to improve computational efficiency by more than a factor of 6 compared to forward-Euler FSM approach. © 2025 The Authors
Mechanics of Advanced Materials and Structures (15210596)31(13)pp. 2844-2858
This study is a first attempt to provide an effective tool for crack prediction in beam-type structures using the peridynamics (PD) theory and metaheuristic-based optimization. The norm of the difference between the response of the tested beam and that of the trial model (analyzed using PD), is minimized to find the position and depth of crack. Model calibration is discussed thoroughly. Five well-known metaheuristics as well as an upgraded charged system search algorithm are considered. Results reveal that the proposed procedure can effectively localize and detect the crack severity, even in noise contaminated cases. © 2023 Taylor & Francis Group, LLC.
Computers and Mathematics with Applications (08981221)136pp. 165-190
In this paper an implicit peridynamic (PD) framework is devised for the solution of quasi-static problems involving brittle and quasi-brittle fracture. A common way to deal with such problems, in explicit frameworks, is to incorporate artificial damping into the PD equation of motion. This may contribute to non-physical behavior of the system. Also, implicit frameworks, based on the Newton-Raphson method, fail to trace the snap-back behavior. To this end, we develop an implicit peridynamic framework to deal with quasi-static problems involving snap-back/snap-through instabilities. This is achieved through two different approaches. The first approach is based on the global arc-length method applicable to problems with zero displacement constraints. The second one is based on a displacement-controlled arc-length method, suitable for problems with non-zero prescribed displacements. The framework is developed for linear elastic as well as softening (degrading) material damage models, respectively appropriate for brittle and quasi-brittle fracture modeling. The robust performance of the proposed framework, within various loading scenarios and damage models, is demonstrated. To boost the numerical performance of the proposed PD framework, a hybrid integration scheme is employed. We demonstrate the advantages of this scheme over the conventional (standard) one in terms of efficiency for quasi-static problems. © 2023 Elsevier Ltd
Theoretical and Applied Fracture Mechanics (01678442)128
This study introduces a bond-based peridynamic (BB-PD) approach for analyzing concrete structures under cyclic loading, with the aim of comprehensively addressing crack propagation and concrete crushing phenomena. The model's generality is enhanced by eliminating the need for calibration. Additionally, a practical method is proposed to incorporate the hardening effect of steel bars, making it applicable to reinforced concrete structures. The accuracy of the approach is evaluated through numerical examples under various loading conditions, and the results are validated by comparing them with standard experiments using an implicit arc-length method. By eliminating the use of constant or adaptive parameters typically found in explicit methods, the arc-length method, combined with the proposed material model, effectively captures the behavior of both ordinary and reinforced concretes subjected to cyclic loading. © 2023 Elsevier Ltd
Engineering with Computers (14355663)39(4)pp. 2807-2828
Coupling of methods based on the classical continuum mechanics (CCM), with peridynamic (PD) models is a recent hot topic in the realm of computational mechanics. In the coupled models, to optimize the usage of computational resources, usually the application of PD (the more demanding procedure) is restricted to critical areas of the domain affected by discontinuities such as propagating cracks. The remaining parts of the domain are described by a more efficient CCM-based model such as the finite element method (FEM). Here, we develop a coupled FEM/PD model for dynamic fracture modeling. The proposed method simultaneously features the following: (1) it can adaptively change the coupling configuration throughout the simulation such that only critical zones, on the verge of crack nucleation/propagation, are tackled by the PD procedure, and (2) it appropriately supports different grid spacing of PD and FEM parts. We refer to a model possessing both the features as multi-adaptive. This is crucial for a highly efficient coupling scheme. The performance of the proposed method is analyzed in terms of accuracy and computational efficiency through different numerical examples. The results show that the proposed method is superior to using a refined PD model, since it provides the same level of accuracy at a much lower computational cost. As a novel application, we present the promising results of a crack propagation problem in an unbounded domain, solved using classical artificial boundary conditions on an outer FEM layer. © 2022, The Author(s), under exclusive licence to Springer-Verlag London Ltd., part of Springer Nature.
Ongaro, G.,
Shojaei a., A.,
Mossaiby, F.,
Hermann, A.,
Cyron, C.,
Trovalusci, P. International Journal Of Fracture (15732673)244(1-2)pp. 1-24
Peridynamic (PD) models are commonly implemented by exploiting a particle-based method referred to as standard scheme. Compared to numerical methods based on classical theories (e.g., the finite element method), PD models using the meshfree standard scheme are typically computationally more expensive mainly for two reasons. First, the nonlocal nature of PD requires advanced quadrature schemes. Second, non-uniform discretizations of the standard scheme are inaccurate and thus typically avoided. Hence, very fine uniform discretizations are applied in the whole domain even in cases where a fine resolution is per se required only in a small part of it (e.g., close to discontinuities and interfaces). In the present study, a new framework is devised to enhance the computational performance of PD models substantially. It applies the standard scheme only to localized regions where discontinuities and interfaces emerge, and a less demanding quadrature scheme to the rest of the domain. Moreover, it uses a multi-grid approach with a fine grid spacing only in critical regions. Because these regions are identified dynamically over time, our framework is referred to as multi-adaptive. The performance of the proposed approach is examined by means of two real-world problems, the Kalthoff–Winkler experiment and the bio-degradation of a magnesium-based bone implant screw. It is demonstrated that our novel framework can vastly reduce the computational cost (for given accuracy requirements) compared to a simple application of the standard scheme. © 2023, The Author(s).
Computers and Mathematics with Applications (08981221)124pp. 163-187
Transient nonlinear problems play an important role in many engineering problems. Phase-field equations, including the well-known Allen-Cahn and Cahn-Hilliard equations, fall in this category, and have applications in cutting-edge technologies such as modeling the diffusion of lithium (Li) ions in two-phase electrode particles of Li-ion batteries. In this paper, a local meshless method for solving this category of partial differential equations (PDEs) is proposed. The Newton-Kantorovich scheme is employed to transform the nonlinear PDEs to an iterative series of linear ones which can be solved with the proposed method. The accuracy and performance of the method are examined in various linear and nonlinear problems, such as Laplace equation, three dimensional elasticity as well as some abstract mathematical equations with linear or nonlinear boundary conditions. The main focus of the work is on applying the proposed method in solution of the phase-field equations, including the Allen-Cahn and Cahn-Hilliard equations. In addition to homogeneous Neumann boundary condition which has been widely examined in the literature, we also employ a practical nonlinear, inhomogeneous Neumann boundary condition formulation specialized for modeling the diffusion of lithium ions in electrode particles of Li-ion batteries. The generalized-α method is used for time integration of diffusion-type equations to overcome the intrinsic stiffness of the phase-field equations. It is shown that the method is capable of capturing the main features of the phase-field models i.e. phase separation, coarsening and energy decay in closed systems. © 2022 Elsevier Ltd
Meccanica (15729648)57(10)pp. 2517-2531
In this paper we present a new approach based on peridynamics (PD) for identification of crack position and length in membranes, as one of the most important topics in structural health monitoring. PD is known to provide rather realistic results in problems involving crack nucleation and propagation and requires no special treatment for consideration of cracks; therefore it is employed in this work for computation of in-plane response of the cracked membranes under impact load conditions. To identify the crack characteristics, the difference between measured and computed maximum displacements of some predetermined points is considered as the objective function. An optimization step, using the Charged System Search (CSS) algorithm, is then used to minimize the objective function. To increase the efficiency of CSS, the algorithm is upgraded using a sinusoidal search phase. To evaluate the proposed approach, four scenarios of crack placement are considered. Noisy conditions are also considered in each scenario. The results demonstrate the efficiency of the proposed approach in determining the position, orientation, and length of the crack. Also, the upgraded CSS algorithm consistently outperforms the original CSS algorithm. © 2022, Springer Nature B.V.
International Journal of Mechanical Sciences (00207403)194
Phase separation arises in many materials systems as a result of solute intercalation. It is in particular known that the mechanical stresses resulting from phase separation in lithium-ion batteries can be large enough to cause formation of a variety of defects and degradation of the host electrode upon cycling. Fracture mechanics models have been previously developed for identifying the critical conditions which lead to the growth of a pre-existing crack in two-phase electrode particles. Relying on a cohesive zone model in combination with the distributed dislocation technique, this work examines critical conditions corresponding to the onset of crack formation in an initially crack-free two-phase electrode. Considering a phase separating planar electrode, we utilize a Cahn-Hilliard type phase field model for capturing evolution of the concentration profiles during both intercalation and deintercalation half-cycles. Crack formation in the electrode subject to the diffusion-induced stresses is considered to result from strain localization at the place of pre-existing defects or weaknesses in the material whose behavior is modelled using a linearly softening traction-separation law. Numerical solution of the governing equations allows identification of a flaw-tolerant electrode thickness below which crack formation in the electrode becomes suppressed in the sense that the maximum opening along the cohesive zone cannot reach the critical separation required for crack formation, and hence, failure in the electrode is predicted to be dominated by the theoretical strength of the material rather than by crack formation. Since in the limit of small surface fluxes, uniform axial stresses develop in the individual phases, results of the analyses are also examined with reference to the results which follow from the analysis of the planar structure subject to uniform tension. © 2020
Engineering Fracture Mechanics (00137944)223
A thermo-mechanical peridynamic model using adaptive grid refinement is developed to investigate crack propagation in ceramics. Compared to a standard peridynamic model, using uniform grid, this approach allows to increase the resolution of analysis only in the critical zones. The performance of this approach in solving 2D thermo-elastic problems is examined and then it is applied to study the fracture of a ceramic disk under central thermal shock. Finally, the proposed approach is adopted to investigate thermal shock in thin rectangular and circular slabs. The accuracy of the method is evaluated by comparing its numerical results with those obtained by applying the finite element method (FEM), a standard peridynamic approach or with experimental data available in the literature. A proper agreement is achieved at a smaller computational cost. © 2019 Elsevier Ltd
Mossaiby, F.,
Shojaei a., A.,
Boroomand b., B.,
Zaccariotto, M.,
Galvanetto, U. Computer Methods in Applied Mechanics and Engineering (00457825)362
In this paper a new collocation technique for constructing time-dependent absorbing boundary conditions (ABCs) applicable to elastic wave motion is devised. The approach makes use of plane waves which satisfy the governing equations of motion to construct the absorbing boundary conditions. The plane waves are adjusted so that they can cope with the satisfaction of radiation boundary conditions. The proposed technique offers some advantages and exhibits the following features: it is easy to implement; its approximation scheme is local in space and time and thus it does not deal with any routine schemes such as Fourier and Laplace transform, making the method computationally less demanding; as the employed basis functions used to construct the absorbing boundary condition are residual-free, it requires neither any differential operator (to approximate the wave dispersion relation), nor any auxiliary variables; it constructs Dirichlet-type ABCs and hence no derivatives of the field variables are required for the imposition of radiation conditions. In this study, we apply the proposed technique to the solution procedure of a collocation approach based on the finite point method which proceeds in time by an explicit velocity-Verlet algorithm. It contributes to developing a consistent meshless framework for the solution of unbounded elastodynamic problems in time domain. We also apply the proposed method to a standard finite element solver. The performance of the method in solution of some 2D examples is examined. We shall show that the method exhibits appropriate results, conserves the energy almost exactly, and it performs stably in time even in the case of long-term computations. © 2020 Elsevier B.V.
Journal of Water Process Engineering (22147144)33
Numerous pilot and computational studies have investigated effects of inlet, outlet, baffles and geometry on the hydraulic performance of sedimentation ponds in water and wastewater treatment plants. However, no full-scale study has addressed buoyancy effects caused by variations in the ambient temperature and the real-time temperature of the fluid at the inlet. This might significantly affect the hydraulic performance of sedimentation ponds. For the first time, we conduct representative simulations to study effects of variations in the temperature on the hydraulic performance of a plant-scale rectangular pond. It is shown that even a small difference between the temperature at the inlet and the bulk flow inside the pond could augment thermal stratification, as a result of buoyancy forces. The results reveal that buoyancy forces in a typical pond would decrease the hydraulic retention time and the effective volume. We suggest that understanding the effect of temperature profiles on the hydraulic performance of ponds should be a prerequisite for improving the design and configuration. © 2019 Elsevier Ltd
Shojaei a., A.,
Mossaiby, F.,
Zaccariotto, M.,
Galvanetto, U. Computer Methods in Applied Mechanics and Engineering (00457825)356pp. 629-651
This paper introduces an effective way to equip the standard finite element method (FEM) for the solution of transient scalar wave propagation problems in unbounded domains. Similar to many other methods, we truncate the unbounded domain at an artificial boundary and convert the problem into a bounded one by prescribing appropriate absorbing boundary conditions (ABCs) at the truncating boundary. In the present method, the ABCs are time-dependent, and they are constructed by a simple collocation approach which is local in space and time. Therefore, the method does not make use of any routine schemes such as Fourier and Laplace transform. We shall show that the method is simple, and it can be easily applied to an explicit time domain FEM approach so that the sparsity of the FEM scheme (as well as its efficiency) can be preserved. The proposed method does not require any auxiliary variables as well as any approximating differential operators. This feature roots from the fact that here the ABCs are Dirichlet-type (or first-type) and thus they can be easily imposed to the corresponding boundaries. Therefore, the method shares some similarities with the conventional 1st and 2nd order ABC methods in terms of the simplicity of implementation. The method employs basis functions that exactly satisfy the governing and dispersion wave equations. The basis function can be easily adjusted to act as outgoing waves transmitting energy from the interior domain (near field) towards exterior domain (far field); i.e, they can cope with satisfaction of radiation boundary conditions. Several numerical examples are presented to evaluate the performance and to demonstrate the effectiveness of the approach. We shall show that the present method is capable of yielding results with a proper level of accuracy, similar to that of the perfectly matched layers method (PMLs), and it performs stably even in the case of long-term computations. © 2019 Elsevier B.V.
European Journal of Mechanics, A/Solids (09977538)74pp. 96-111
Phase separation has been widely observed in various energy storage systems, and is known to severely impact mechanical integrity and electrochemical performance in lithium intercalation materials. Core-shell type models have been extensively used for modeling diffusion, phase transformation, deformation, and stress generation in phase changing energy storage materials. The purpose of this work is to present a systematic fracture mechanics analysis of two-phase electrode particles with core-shell structure subject to deintercalation using both numerical and analytical models. Mechanical behavior of the host solid is assumed to follow small deformation linear elasticity, and geometry of the particles is considered to be cylindrical/spherical. A moving interphase model accounting for the effect of solution non-ideality is utilized in combination with the weight function method of fracture mechanics to numerically calculate stress intensity factors (SIFs) for pre-existing cracks at the particle surface. Further, an analytical solution in the limit of small surface fluxes and a semi-analytical solution in the presence of large surface fluxes are also developed for the maximum SIF which could arise for a pre-existing surface crack during a complete deintercalation half-cycle. Implication of the results in terms of prediction of a critical particle size to avoid fracture in two-phase electrode particles is also presented. Numerical results along with their comparison with analytical predictions are presented in terms of the concentration and stress profiles, maximum tensile hoop stress, and maximum SIFs for a range of two-phase regular solid solutions and subject to a broad range of surface deintercalation fluxes. We further examine the results of this work by considering a lithium deintercalation system whose thermodynamic behavior is considerably different from that of a regular solution, and the results found using analytical and numerical models are compared. © 2018 Elsevier Masson SAS
Geomatics, Natural Hazards and Risk (19475713)10(1)pp. 797-819
Volcano is one of the geodynamic phenomena causing irreparable damages. As lava accumulates in reservoir and then comes to the surface, geometry of the source can be used to predict volcanic eruptions. In this study, using the inverse method of fundamental solutions (MFS) and taking into account the effect of topography, the geometry of the source including shape, depth and centre position of the magma tank is estimated. The MFS is a numerical method for solving boundary value problems with known partial differential equations. The displacement field calculated in the previous studies using InSAR for deflation mode of Cerro Blanco volcano was utilized in this study. It was estimated that the magma source of the volcano is a sphere with a radius of 1 km located at a horizontal position of (1:0016±4e-6;-1:0016±1e-6) km and the depth of about 10 km from the summit with respect to the defined coordinate system. This finding is consistent with that of recent studies in which inversion of InSAR data was used to analyse the geometry of the magma source. The RMSE between the deformation fields of the magma source calculated in the previous studies and that of the study herein via MFS was approximately 3 mm. © 2019, © 2019 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
Computational Mechanics (14320924)63(5)pp. 805-819
Efficient simulation of wave propagation in heterogeneous materials is still a challenging task. The spectral cell method, representing a combination of spectral elements with the fictitious domain concept, has proven to be an efficient approach for wave propagation analysis in materials with complicated microstructure. In this paper, we report details of parallel implementation of the spectral cell method using multi-core CPUs as well as GPUs. In our CPU implementation, we employ the OpenMP directives to parallelize the loops. On GPUs, however, we use the OpenCL framework to develop single- and multi-GPU versions of the code. In all of our implementations, the core operation is a sparse matrix-vector multiplication (SpMV) kernel. We analyze each implementation to determine its features and bottlenecks. The results show that speedups of up to 128 relative to serial CPU code can be achieved using multi-GPU code. © 2018, Springer-Verlag GmbH Germany, part of Springer Nature.
Shojaei a., A.,
Mossaiby, F.,
Zaccariotto, M.,
Galvanetto, U. International Journal of Mechanical Sciences (00207403)144pp. 600-617
The standard way of implementing Peridynamics is a meshfree approach which uses a uniform discretization. This is inefficient when a very dense grid spacing for a localized area is required. In this paper, a radically new strategy to couple grids with different spacing is put forward. It is free of ghost forces in static cases and spurious waves in dynamic problems can be controlled and made negligible thanks to proper discretization. There is no loss of volume due to non-uniform discretization at the interface between different grids. An efficient algorithm is developed to apply the refinement adaptively. It permits to increase the resolution of the analysis only in the critical zones. The performance is investigated by solving dynamic problems, including cases of crack propagation in brittle materials. We compare the solutions of the proposed method with those of a standard peridynamic model, which employs uniform discretization, and show that the same accuracy is obtained at a much smaller computational cost. © 2018 Elsevier Ltd
Journal of Power Sources (03787753)350pp. 127-139
It is well known that phase separation could severely intensify mechanical degradation and expedite capacity fading in lithium-ion battery electrodes during electrochemical cycling. Experiments have frequently revealed that such degradation effects could be substantially mitigated via reducing the electrode feature size to the nanoscale. The purpose of this work is to present a fracture mechanics study of the phase separating planar electrodes. To this end, a phase field model is utilized to predict how phase separation affects evolution of the solute distribution and stress profile in a planar electrode. Behavior of the preexisting flaws in the electrode in response to the diffusion induced stresses is then examined via computing the time dependent stress intensity factor arising at the tip of flaws during both the insertion and extraction half-cycles. Further, adopting a sharp-interphase approximation of the system, a critical electrode thickness is derived below which the phase separating electrode becomes flaw tolerant. Numerical results of the phase field model are also compared against analytical predictions of the sharp-interphase model. The results are further discussed with reference to the available experiments in the literature. Finally, some of the limitations of the model are cautioned. © 2017 Elsevier B.V.
Geomatics, Natural Hazards and Risk (19475713)8(2)pp. 1258-1275
According to geodetic research works, surface deformation in the form of uplift or subsidence in volcanic areas is either a sign of magma moving towards the opening of the volcano (inflation) or removal of the magma source (deflation). Using the new method of fundamental solutions (MFS) in this study, a deformation field for the surface of the volcano is calculated considering the effect of topography. MFS is a numerical technique for solving boundary value problems with known partial differential equations. This technique has not been used in volcanic deformation studies so far. Because of the simplicity and efficiency of the technique, it is also an effective tool for solving a wide range of problems in other fields of science and technology. To test the method, the displacement calculated using the MFS was compared with that of the interferometric synthetic aperture radar observations from the previous study in Cerro Blanco volcano. The volcano was in the deflation mode during this period at the rate of 1.2 cm/yr. The comparison showed a root-mean-square error (RMSE) in the order of 2 mm which represents a satisfactory agreement with the results of the observations, less than the RMSE of the analytical models considered. © 2017 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group.
Mossaiby, F.,
Shojaei a., A.,
Zaccariotto, M.,
Galvanetto, U. Computers and Mathematics with Applications (08981221)74(8)pp. 1856-1870
Parallel processing is one of the major trends in the computational mechanics community. Due to inherent limitations in processor design, manufacturers have shifted towards the multi- and many-core architectures. The graphics processing units (GPUs) are gaining more and more popularity due to high availability and processing power as well as maturity of development tools and community experience. In this research we describe a rather general approach to using OpenCL implementation of 3D Peridynamics model on GPU platform. Peridynamics is a non-local continuum theory for describing the behavior of material used especially when damage and crack nucleation or propagation is of interest. The steps taken for developing an OpenMP code from the serial one as well as the comparison between OpenCL and OpenMP codes are provided. Optimization techniques and their effects on the performance of the code are described. The implementations are tested on some 3D benchmarks with hundred of thousands to millions of nodes. The behavior of codes in terms of being memory or compute bound are analyzed. In all test cases reported, the OpenCL implementation consistently outperforms serial and OpenMP ones and paves the road for the development of high performance Peridynamics codes. © 2017 Elsevier Ltd
Shojaei a., A.,
Mossaiby, F.,
Zaccariotto, M.,
Galvanetto, U. Acta Mechanica (16196937)228(10)pp. 3581-3593
In this paper, the application of the meshless finite point method (FPM) to solve elastodynamic problems through an explicit velocity–Verlet time integration method is investigated. Strong form-based methods, such as the FPM, are generally less stable and accurate in terms of satisfaction of Neumann boundary conditions than weak form-based methods. This is due to the fact that in such types of methods, Neumann boundary conditions must be imposed by a series of equations which are different from the governing equations in the problem domain. In this paper, keeping all the advantages of FPM in terms of simplicity and efficiency, a new simple strategy for proper satisfaction of Neumann boundary conditions in time for elastodynamic problems is investigated. The method is described in detail, and several numerical examples are presented. Moreover, the accuracy of the method with reference to the solution of some 3D problems is discussed. © 2017, Springer-Verlag GmbH Austria.
Engineering Computations (02644401)33(8)pp. 2238-2263
Purpose - The purpose of this paper is to extend the meshless local exponential basis functions (MLEBF) method to the case of nonlinear and linear, variable coefficient partial differential equations (PDEs). Design/methodology/approach - The original version of MLEBF method is limited to linear, constant coefficient PDEs. The reason is that exponential bases which satisfy the homogeneous operator can only be determined for this class of problems. To extend this method to the general case of linear PDEs, the variable coefficients along with all involved derivatives are first expanded. This expanded form is evaluated at the center of each cloud, and is assumed to be constant over the entire cloud. The solution procedure is followed as in the former version. Nonlinear problems are first converted to a succession of linear, variable coefficient PDEs using the Newton-Kantorovich scheme and are subsequently solved using the aforementioned approach until convergence is achieved. Findings - The results obtained show good performance of the method as solution to a wide range of problems. The results are compared with the well-known methods in the literature such as the finite element method, high-order finite difference method or variants of the boundary element method. Originality/value - The MLEBF method is a simple yet effective tool for analyzing various kinds of problems. It is easy to implement with high parallelization potential. The proposed method addresses the biggest limitation of the method, and extends it to linear, variable coefficient PDEs as well as nonlinear ones. © 2016 Emerald Group Publishing Limited.
Scripta Materialia (13596462)114pp. 142-145
In the past few decades, nanostructures of ceria have received significant attention for potential applications in a variety of technologies. Experiments have frequently shown that physical properties of ceria nanomaterials are strongly size-dependent. Here, by accounting for the highly non-linear coupling between electrical, chemical and mechanical driving forces, we develop a continuum model which allows us to investigate equilibrium distribution of defects in nanowires and nanotubes of ceria. It is shown that the model predicts strongly size-dependent non-stoichiometric composition, lattice constant and surface stresses in ceria nanostructures whose diameter is comparable with the Debye's length scale of the problem. © 2015 Elsevier B.V. All rights reserved.
Journal of the Mechanics and Physics of Solids (00225096)88pp. 1-11
Owing to its broad potential applications, nanostructured ceria has been subject of intense investigation in the past few decades. Experiments have demonstrated that various material properties of the nanostructured ionic solids including ceria vary with the feature size. Here, we present a theoretical study of the size effects on the composition, defect concentrations and stresses in free-standing nanoparticles of nonstoichiometric ionic solids. To this end, a continuum model is developed which accounts for the highly nonlinear coupling between mechanical, chemical and electrical driving forces, and their effects on the thermodynamic equilibrium of the defect species. It is demonstrated that the model, once applied to the case of ceria, predicts size-dependent defect concentrations and surface stresses. It is further shown that the theoretical predictions of the size effects on the composition and lattice parameter are in good agreement with the experimental observations. © 2015 Elsevier Ltd. All rights reserved.
International Journal for Numerical Methods in Engineering (00295981)105(3)pp. 221-240
In this paper, we address shortcomings of the method of exponential basis functions by extending it to general linear and non-linear problems. In linear problems, the solution is approximated using a linear combination of exponential functions. The coefficients are calculated such that the homogenous form of equation is satisfied on some grid. To solve non-linear problems, they are converted to into a succession of linear ones using a Newton-Kantorovich approach. The generalized exponential basis functions (GEBF) method developed can be implemented with greater ease compared with exponential basis functions, as all calculations can be performed using real numbers and no characteristic equation is needed. The details of an optimized implementation are described. We compare GEBF on some benchmark problems with methods in the literature, such as variants of the boundary element method, where GEBF shows a good performance. Also, in a 3D problem, we report the run time of the proposed method compared with that of Kratos, a parallel, highly optimized finite element code. The results show that in this example, to obtain the same level of error, much less computational effort is needed in the proposed method. Practical limitations might be encountered, however, for large problems because of dense matrix operations involved. © 2016 John Wiley and Sons, Ltd.
Engineering Computations (02644401)32(6)pp. 1567-1600
Purpose - The purpose of this paper is to present a simple meshless solution method for challenging engineering problems such as those with high wave numbers or convection-diffusion ones with high Peclet number. The method uses a set of residual-free bases in a local form. Design/methodology/approach - The residual-free bases, called here as exponential basis functions, are found so that they satisfy the governing equations within each subdomain. The compatibility between the subdomains is weakly satisfied by enforcing the local approximation of the main state variables pass through the data at nodes surrounding the central node of the subdomain. The central state variable is first recovered from the approximation and then re-assigned to the central node to construct the associated equation. This leads to the least compatibility required in the solution, e.g. C0 continuity in Laplace problems. Findings - The authors shall show that one can solve a variety of problems with regular and irregular point distribution with high convergence rate. The authors demonstrate that this is impossible to achieve using finite element method. Problems with Laplace and Helmholtz operators as well as elasto-static problems are solved to demonstrate the effectiveness of the method. A convection-diffusion problem with high Peclet number and problems with high wave numbers are among the examples. In all cases, results with high rate of convergence are obtained with moderate number of nodes per cloud. Originality/value - The paper presents a simple meshless method which not only is capable of solving a variety of challenging engineering problems but also yields results with high convergence rate. © Emerald Group Publishing Limited.
Engineering Computations (02644401)32(2)pp. 406-423
Purpose: The purpose of this paper is to aim at extending the method of exponential basis functions (EBF) to solve a class of problems with singularities. Design/methodology/approach: In the procedure of EBF a summation of EBF satisfying the governing differential equation with unknown constant coefficients is considered for the solution. These coefficients are determined by the satisfaction of prescribed boundary conditions through a collocation approach. The applied basis functions are available in the case of linear partial differential equations (PDEs) with constant coefficients. Moreover, the method contributes to yield highly accurate results with ultra convergence rates for problems with smooth solution. This leads EBF to offer many advantages for a variety of engineering problems. However, owing to the global and smooth nature of the bases, the performance of EBF deteriorates in problems with singularities. In the present study, some exponential-like influence functions are developed, and a few of them are added to original bases. Findings: The new bases are capable of forming the constitutive terms of the asymptotic solution near the singularity points and alleviate the aforementioned limitation. The appealing feature of this method is that all the advantages of EBF such as its simplicity and efficiency are completely preserved. Research limitations/implications: In its current form, EBF can only solve PDEs with constant coefficients. Originality/value: Application of the method to some benchmark problems demonstrates its robustness over some other boundary approximation methods. This research may pave the road for future investigations corresponding to a wide range of practical engineering problems. © Emerald Group Publishing Limited.
Computational Mechanics (14320924)53(6)pp. 1355-1374
This paper introduces a novel meshless method based on the local use of exponential basis functions (EBFs). The EBFs are found so that they satisfy the governing equations within a series of subdomains. The compatibility between the subdomains is weakly satisfied through the minimization of a suitable norm written for the residuals of the continuity conditions. The residual norm may contain any desirable order of continuity. This allows increasing the continuity of the solution without increasing the type of point-wise variables at each node. The solution procedure begins with the discretization of the solution domain into a set of nodal points and cloud construction on each nodal point. The approximate solution in the local coordinates of each cloud is constructed by a series of EBFs. A set of intermediate points are distributed throughout the domain and its boundary to apply the continuity between the local solutions of the adjacent clouds up to a desired order, and also to impose the boundary conditions. The main nodes may play the role of the intermediate points as well. The validity of the results is shown through some patch tests. Also some numerical examples are solved to illustrate the capabilities of the method. High convergence rate of the numerical results is one of the salient features of the proposed meshless method. © 2014 Springer-Verlag Berlin Heidelberg.
Computers and Fluids (00457930)81pp. 134-144
The rise of GPUs in modern high-performance systems increases the interest in porting portion of codes to such hardware. The current paper aims to explore the performance of a portable state-of-the-art FE solver on GPU accelerators. Performance evaluation is done by comparing with an existing highly-optimized OpenMP version of the solver. Code portability is ensured by writing the program using the OpenCL 1.1 specifications, while performance portability is sought through an optimization step performed at the beginning of the calculations to find out the optimal parameter set for the solver. The results show that the new implementation can be several times faster than the OpenMP version. © 2013 Elsevier Ltd.
International Journal for Numerical Methods in Engineering (00295981)89(13)pp. 1635-1651
The solution of problems in computational fluid dynamics (CFD) represents a classical field for the application of advanced numerical methods. Many different approaches were developed over the years to address CFD applications. Good examples are finite volumes, finite differences (FD), and finite elements (FE) but also newer approaches such as the lattice-Boltzmann (LB), smooth particle hydrodynamics or the particle finite element method. FD and LB methods on regular grids are known to be superior in terms of raw computing speed, but using such regular discretization represents an important limitation in dealing with complex geometries. Here, we concentrate on unstructured approaches which are less common in the GPU world. We employ a nonstandard FE approach which leverages an optimized edge-based data structure allowing a highly parallel implementation. Such technique is applied to the 'convection-diffusion' problem, which is often considered as a first step towards CFD because of similarities to the nonconservative form of the Navier-Stokes equations. In this regard, an existing highly optimized parallel OpenMP solver is ported to graphics hardware based on the OpenCL platform. The optimizations performed are discussed in detail. A number of benchmarks prove that the GPU-accelerated OpenCL code consistently outperforms the OpenMP version. © 2011 John Wiley & Sons, Ltd.
Civil-Comp Proceedings (17593433)83
In this paper we present a numerical method suitable for solution of steady state wave problems in which the material properties vary periodically throughout the domain. Discrete Green's functions, in finite element sense, are selected as the representatives of the problems. The Green's functions are evaluated on unbounded domains and this involves satisfaction of radiation conditions in the solutions. The formulation, given in this paper, is the extension of the one recently proposed by the authors for domains with homogenous materials. Here, the principles of Floquet theory for solution of partial differential equations, with periodic coefficients, are used. First, the fundamental exponential-like wave bases are obtained through the dispersion relations and then the radiation conditions are satisfied by selecting the wave bases. For selecting the wave bases, a quadrant of the main unbounded domain together with a set of appropriate boundary conditions is considered. For satisfaction of the boundary conditions a discrete transformation technique, proposed by the authors, is used. Application of the method is shown on a sample problem and the results are compared with those obtained form exact solution of a rather similar problem with homogenized material. The comparison shows the validity of the solution method. © 2006 Civil-Comp Press.
International Journal for Numerical Methods in Engineering (00295981)67(11)pp. 1491-1530
In this paper we present a new approach for finite element solution of time-harmonic wave problems on unbounded domains. As representatives of the wave problems, discrete Green's functions are evaluated in finite element sense. The finite element mesh is considered to be of repeatable pattern (cell) constructed in rectangular co-ordinates. The system of FE equations is therefore reduced to a set of well-known dispersion equations by using a spectral solution approach. The spectral wave bases are constructed directly from the FE dispersion equations. Radiation condition is satisfied by selecting the wave bases so that the wave information is transmitted in appropriate directions at the cell level. Dirichlet/Neumann boundary conditions are defined at the edges of a quadrant of the main domain while using the axes of symmetry of the problem. A new discrete transformation method, recently proposed by the authors, is used to satisfy the boundary conditions. Comprehensive studies are made for showing the validity, accuracy and convergence of the solutions. The results of the benchmark problems indicate that the proposed method can be used to evaluate discrete Green's functions whose analytical forms are not available. Copyright © 2006 John Wiley & Sons, Ltd.
International Journal for Numerical Methods in Engineering (00295981)64(4)pp. 427-460
In this part of paper we shall extend the formulation proposed by Babuška and co-workers for robustness patch test, for quality assessment of error estimators, to more general cases of patch locations especially in three-dimensional problems. This is performed first by finding an asymptotic finite element solution at interior parts of a problem with assumed smooth exact solution and then adding a correction part to obtain the solution near a kinked boundary irrespective of other boundary conditions at far ends of the domain. It has been shown that the solution corresponding to the correction part may be obtained in a spectral form by assuming a suitable proportionality relation between the nodal values of a mesh with repeatable pattern of macro-patches. Ha ving found the asymptotic finite element solution, the performance of error estimators may be examined. Although in this paper we focus on the asymptotic behaviour of error estimators, the method described in this part may be used to obtain finite element solution for two/ three-dimensional unbounded heat/elasticity problems with homogeneous differential equations. Some numerical results are presented to show the validity and performance of the proposed method. Copyright © 2005 John Wiley & Sons, Ltd.
International Journal for Numerical Methods in Engineering (00295981)64(4)pp. 461-502
In this part of the paper we shall use the formulation given in the first part to assess the quality of recovery-based error estimators using two recovery methods, i.e. superconvergent patch recovery (SPR) and recovery by equilibrium in patches (REP). The recovery methods have been shown to be asymptotically robust and superconvergent when applied to two-dimensional problems. In this study we shall examine the behaviour of the recovery methods on several three-dimensional mesh patterns for patches located either inside or at boundaries. This is performed by first finding an asymptotic finite element solution, irrespective of boundary conditions at far ends of the domain, and then applying the recovery methods. The test procedure near kinked boundaries is explained in a step-by-step manner. The results are given in a series of tables and figures for various cases of three-dimensional mesh patterns. It has been experienced that the full superconvergent property is generally lost due to presence of boundary layer solution and the definition of the recoveries near boundaries though the results of the robustness test is still within an acceptable range. Copyright © 2005 John Wiley & Sons, Ltd.
