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This paper discusses the process of optimizing the shape of systems that are controlled by the Stokes flow with threshold leak boundary conditions. In the theoretical part it focuses on studying the stability of solutions to the state problem in relation to a specific set of domains. In order to facilitate computation, the slip term and impermeability condition are regulated. In the computational part, the optimized portion of the boundary is defined using Bézier polynomials, in order to create a finite dimensional optimization problem. The paper also includes numerical examples to demonstrate the computational efficiency of this approach. [ABSTRACT FROM AUTHOR]

In this paper, we consider an enriched cut finite element method (ECFEM) with interface-unfitted meshes for solving Stokes interface equations consisting of two incompressible fluids with different viscosities. By approximating the velocity with the enriched P 1 element and the pressure with the P 0 element, and stabilizing the Galerkin variational formulation with suitable ghost penalty terms, we propose the new ECFEM and prove that it is well-posed and has the optimal a priori error estimate in the energy norm. All derived results are independent of the interface position. Moreover, compared with other conforming finite element methods with the optimal rate in convergence, the proposed scheme here not only has the minimum degrees of freedom, but also avoids using the derivative of the pressure in the penalty term. The presented numerical examples validate the theoretical predictions. [ABSTRACT FROM AUTHOR]

Models involving interfaces with discontinuities or even singularities of some fields across them are very frequent in real life problems modelling. In the last decades, the use of the eXtended Finite Element Method (XFEM) instead of the traditional FEM has become more and more popular, mainly because of two advantages: the mesh of the domain can be independent of the interface position, therefore avoiding remeshing, and it allows to enrich an area with specific shape functions fitted to the particular properties (singularities, discontinuities) of the expected solution, obtaining more accurate results with less computational efforts. Nevertheless, a critical point of XFEM is its implementation since it varies from one problem to another, due to the different kind (and number) of degrees of freedom on each node. A diligent organization of nodes, degrees of freedom and enrichment functions is fundamental to achieve an efficient implementation. Our aim in this paper is to provide a common reference framework for the implementation of XFEM from a mathematical point of view, providing the readers with a set of tools that will allow them to apply it to any kind of problem. To this aim, we present a detailed description of XFEM implementation, with special emphasis on the terms that involve integration over interfaces. The proposed tools are presented in a general context, and as an example, we will apply them to a problem of solids mechanics. In particular, we will contextualize the procedure on a Rayleigh waves propagation problem in a cracked structure considering a Signorini contact condition on the crack sides. [ABSTRACT FROM AUTHOR]

In this paper, the Theory of Functional Connections (TFC) is used to analyze static beams, accounting for the von Kármán nonlinearity and using the Timoshenko–Ehrenfest beam theory. The authors extend their earlier framework on linear beam bending problems to nonlinear bending problems using TFC. The TFC results and performance parameters are then compared to those of the Finite Element Method (FEM) to both validate the TFC solutions and compare computational efficiencies. In addition, this paper also focuses on the benefits of using TFC over FEM for stress analysis of beam bending problems. Also, a TFC methodology to solve buckling and free vibration problems for the linearized Timoshenko–Ehrenfest beam equations is introduced and validated. The results within this paper suggest that for most static beam bending problems, TFC provides more accurate solutions in terms of the residuals of the differential equations and a faster solution time when compared to the FEM using linear or quadratic approximations. Also, TFC has the added benefit of calculating stress fields that are continuous and smooth everywhere within the domain of the beam while FEM is limited to a piece-wise continuous stress field. [ABSTRACT FROM AUTHOR]

In this paper, a weak Galerkin finite element method is implemented to solve the one-dimensional singularly perturbed reaction–diffusion equation. This weak Galerkin finite element scheme uses piecewise polynomial with degree k ≥ 1 in the interior part of each element and piecewise constant function at the nodes of each element. The existence and uniqueness of the weak Galerkin finite element solution are proved. Based on the interpolation operator and the corresponding approximation properties, an ɛ − uniform error bound of O (N − (k + 1) + ɛ (N − 1 ln N) k) in the energy-like norm is investigated rigorously. Furthermore, an ɛ − uniform error bound of O (N − (k + 1 / 2) + (N − 1 ln N) k) in the balanced norm is established by the weighted local L 2 projection and its corresponding approximation properties. Finally, numerical experiments validate the theoretical results. Moreover, numerical results show that this weak Galerkin finite element solution achieves the convergence rate of O ( (N − 1 ln N) k + 1) in the L 2 norm and the discrete L ∞ norm uniformly with respect to the singular perturbation parameter ɛ. [ABSTRACT FROM AUTHOR]

It is well known that the analytical matrices arising from the discretization of distributed parameter systems using the finite element technique are usually symmetric and banded. How to preserve the coefficient matrices of the updated model being of the same band structure is an important yet difficult challenge for model updating in structural dynamics. In this paper, an iterative method for updating mass, damping and stiffness matrices simultaneously based on partial modal measured data is provided. By the method, the optimal updated matrices can be obtained within finite iteration steps by choosing a special kind of initial matrix triplet. The proposed approach not only preserves the physical connectivity of the original model, but also the updated model reproduces the measured modal data, which can be utilized for various finite element model updating problems. Numerical examples confirm the effectiveness of the introduced method. • An iterative updating method for damped structure systems is established. • The proposed approach preserves the physical connectivity of the original model. • The updated model can reproduce the measured modal data. • The deviation between the updated model and the original model is minimal. [ABSTRACT FROM AUTHOR]

High computational cost in elastoplastic analysis is often handled by the use of high performance parallel computers. However, the presence of both elastic and plastic states leads to the branching issue, which prevents the realization of true parallel performance. The computational efficiency of an elastoplastic analysis using finite element method is largely determined by the performance of repeated solution of linearized system of equations. In this paper, we propose GPU-based matrix-free strategies to compute sparse matrix–vector multiplication (SpMV) in Conjugate Gradient (CG) iterative solver for the acceleration of solution of linear system of equations. Matrix-free solvers never assemble large sparse global tangent matrix and perform the computation directly with small dense elemental matrices, reducing the storage requirement and preventing the use of problematic sparse storage formats. A uniform treatment of elements in elastic and plastic regions is achieved by the proposed single kernel strategy, which prevents branching, avoids redundant computation and provides efficient memory access. In addition, we propose node-based and degrees-of-freedom (DOF)-based parallel strategies for effective implementation of matrix-free SpMV on a GPU. The proposed strategies use single elemental tangent matrix for all elements in elastic region and individual tangent matrices in plastic region. The computational experiments over three large-scale benchmark examples of elastoplasticity reveal that the performance of the node-based and DOF-based parallelization strategies depends on the amount of plasticity in the body. At low plasticity levels, node-based strategy performs best, achieving 3.2 × speedup over an existing GPU-based matrix-free SpMV strategy in the literature. For moderate to high amount of plasticity, the DOF-based strategy outperforms every other strategy and obtains speedups of up to 3.5 × over the existing SpMV strategy. [ABSTRACT FROM AUTHOR]

In this paper, we consider the numerical approximation for a diffuse interface model of the two-phase incompressible inductionless magnetohydrodynamics (MHD) problem. This model consists of Cahn–Hilliard equations, Navier–Stokes equations and Poisson equation. We propose a linear and decoupled finite element method to solve this highly nonlinear and multi-physics system. For the time variable, the discretization is a combination of the first order Euler semi-implicit scheme, several first-order stabilization terms and implicit–explicit treatments for coupling terms. For the space variables, we adopt the finite element discretization. Especially, we approximate the current density and electric potential by inf–sup stable face-volume mixed finite element pairs. With these techniques, the scheme only involves a sequence of decoupled linear equations to solve at each time step. We show that the scheme is provably mass-conservative, charge-conservative and unconditionally energy stable. Numerical experiments are performed to illustrate the properties, accuracy and efficiency of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we study the Galerkin finite element approximation for the space Hadamard fractional partial differential equation. We first introduce a modified Fourier transform to analyse the Hadamard fractional calculus, construct the fractional derivative spaces and fractional Sobolev space. Furthermore, we investigate the existence and uniqueness of the weak solution in the fractional Sobolev space. Then using a newly defined log-Lagrangian polynomial as shape function, we discuss the convergence analysis of the semi-discrete scheme. Together with the Crank–Nicolson scheme in time, we present a fully discrete scheme, analyse the stability and convergence. Finally a numerical example is displayed which support the theoretical analysis. [ABSTRACT FROM AUTHOR]

One way of improving the behavior of finite element schemes for classical, time-dependent Maxwell's equations is to render their hyperbolic character to elliptic form. This paper is devoted to the study of a stabilized linear, domain decomposition, finite element method for the time harmonic Maxwell's equations, in a dual form, obtained through the Laplace transformation in time. The model problem is for the particular case of the dielectric permittivity function which is assumed to be constant in a boundary neighborhood. The discrete problem is coercive in a symmetrized norm, equivalent to the discrete norm of the model problem. This yields discrete stability, which together with continuity guarantees the well-posedness of the discrete problem, cf Arnold et al. (2002) [3] , Di Pietro and Ern (2012) [45]. The convergence is addressed both in a priori and a posteriori settings. In the a priori error estimates we confirm the theoretical convergence of the scheme in a L 2 -based, gradient dependent, triple norm. The order of convergence is O (h) in weighted Sobolev space H w 2 (Ω) , and hence optimal. Here, the weight w ≔ w (ɛ , s) where ɛ is the dielectric permittivity function and s is the Laplace transformation variable. We also derive, similar, optimal a posteriori error estimates controlled by a certain, weighted, norm of the residuals of the computed solution over the domain and at the boundary (involving the relevant jump terns) and hence independent of the unknown exact solution. The a posteriori approach is used, e.g. in constructing adaptive algorithms for the computational purposes, which is the subject of a forthcoming paper. Finally, through implementing several numerical examples, we validate the robustness of the proposed scheme. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the numerical analysis of Fitzhugh–Nagumo (FHN) reaction–diffusion equations. The properties of numerical solutions of a semi-discrete and fully-practical piecewise linear finite element technique are provided. Moreover, for a semi-discrete and fully discrete finite element approximation, we establish a priori estimates and error bounds. We also introduce the results of some numerical examples in one and two dimensions, which confirm the theoretical findings of this paper. [ABSTRACT FROM AUTHOR]

Numerical simulation is more and more used during the design stage of a manufactured product in order to optimize its performances. However, it is often too time consuming, particularly when it is used to solve optimization problems, preventing an intensive usage. A-posteriori Model Order Reduction methods can be very effective to shorten the computational time. An approximated solution is then sought in a space of small dimension defined by a reduced basis. The accuracy of such methods is highly dependent on the choice of the reduced basis, extracted from preliminary numerical simulation. The method usually applied to construct such reduced basis is based on the Singular Value Decomposition (SVD), which can be time consuming, and is not adapted when a large collection of preliminary numerical simulations must be used to construct the basis. An alternative to this approach has been proposed recently with the Maximum Entropy Snapshot Sampling (MESS) method. In this paper, we propose to compare these methods with other approaches usually used for clustering or data classification based on vectors distance calculation, like the Centroidal Voronoi Tessellation (CVT), Density Based Spatial Clustering of Applications with Noise (DBSCAN), and Clustering Using Representatives (CURE). The methods are compared on a complex and realistic nonlinear problem in low frequency electromagnetics. The quality of the reduced bases obtained by the different methods are compared. Then, field distributions and global quantities, like eddy current losses and magnetic energy, are computed from the reconstructed results, to further analyze the quality of the reduced bases. • Description of a posteriori model order reduction methods using projection in a reduced basis. • Singular Value Decomposition, Centroidal Voronoi Tessellation, and 3 other methods. • Application on results from a non-linear low frequency electromagnetic simulation. • Calculation of fields and global values from reconstructed results. • Comparison of studied approaches in terms of projection error. [ABSTRACT FROM AUTHOR]

This paper considers a stabilizer-free weak Galerkin (SFWG) finite element method for the time-dependent convection diffusion reaction equation. We describe error estimate for both semidiscrete and fully discrete schemes and achieve the supercloseness convergence rate, which is two orders higher than the optimal order associated with SFWG finite element space (P k (K) , P k + 1 (∂ K) , [ P k + 1 (K) ] 2). More precisely, we obtain O (h k + 2 + τ 2) in L ∞ (H 1) norm and O (h k + 3 + τ 2) in L ∞ (L 2) norm. Numerous numerical examples are provided to confirm the theoretical findings and efficiency of the proposed method. [ABSTRACT FROM AUTHOR]

In this paper, we study the accuracy and robustness of the computational time reversal approach based on the explicit finite element method for application in nondestructive testing in solids. The main goal of this paper is to find a methodology for correct and accurate reconstruction of the original source time history. For numerical modeling of frontal (forward) and reverse (backward) problems of elastic wave propagation, we use the finite element method and explicit time integration with the lumped mass matrix. The suggested methodology is applicable in each finite element open source or commercial software. A special attention is paid to prescription of boundary conditions/loading for the reverse problem for accurate reconstruction of time history of the original source. For evaluation of the reconstruction quality, we suggest certain cost functions. Based on several numerical tests, we show effects of prescription of boundary conditions/loading in time reversal, effect of mesh size and time step size, an unknown obstacle, a number of sources, and environmental disturbance (noise) on the correctness of reconstruction of the original source. [ABSTRACT FROM AUTHOR]

This paper presents a novel method for demagnetization fault detection in permanent magnet synchronous machine (PMSM). Demagnetization fault is a common problem existing in PMSM which deteriorates the machine performance and can lead to serious subsequential machine damages. Unlike the most popular method based on the analysis of the machine currents for detecting this type of fault, the approach proposed in this paper analyzes the voltage signal of a non-invasive search coil by means of Hilbert Huang transform. Simulations are carried out with a two dimensional finite element analysis (FEA) for different demagnetization levels to test the capability of the proposed method to detect PM demagnetization faults. [ABSTRACT FROM AUTHOR]

The time-fractional coupled Korteweg–de Vries equations (TFCKdVEs) describe various interesting real-world phenomena including wave propagation and the description of shallow water waves on a viscous fluid. This paper presents an accurate and robust numerical technique to solve the TFCKdVE. The cubic B-spline is introduced as a basis function and a quadratic B-spline is used as a test function in a finite element method (FEM) is known as Petrov–Galerkin method. The temporal fractional part is simplified via L 1 formula, while the B-spline is employed for the space approximation. The Lax–Richtmyer stability criterion is applied to analyze the stability of the proposed scheme. Four test problems are solved to check performance and validation of the scheme. The accuracy and efficiency of the proposed method are checked via various error norms. The obtained results show good agreement with the exact solutions and earlier work available in the literature. [ABSTRACT FROM AUTHOR]

This paper considers a two-dimensional tempered time–space fractional diffusion equation with a reaction term on convex domains. Firstly, the analytical solution to a one-dimensional tempered time–space diffusion equation is derived in terms of the Fox H function. However, for a two-dimensional counterpart, an explicit expression for its analytical solution does not seem tractable. This motivates us to resort to numerical methods. Next, the L1 formula on a graded mesh is modified to approximate the Caputo tempered time-fractional derivative. A fast evaluation for the tempered time-fractional operator is developed based on a sum-of-exponentials approximation, reducing the computational work and storage significantly. Furthermore, the Galerkin finite element method based on an unstructured mesh is utilised to solve the problem. Its stability and convergence are established. Finally, two numerical examples in different convex domains are investigated to demonstrate the effectiveness of the numerical method. As an application, the tempered fractional Bloch–Torrey equation retaining Larmor precession in a human brain-like domain is illustrated to observe the evolution of the transverse magnetisation. An interesting finding is that the tempered parameter has a significant impact on the decay of magnetisation. [ABSTRACT FROM AUTHOR]

In this paper, a Nitsche mixed extended finite element method based on Ciarlet–Raviart formulation is proposed to discretize the biharmonic interface problem with interface unfitted meshes. We present a discrete mixed approximate formulation based on the piecewise quadratic continuous finite element space. By adding a stabilization procedure, we get the discrete inf–sup condition and prove a suboptimal a priori error estimate for the discrete problem. If the solution of the dual problem satisfies the additional regularity condition, then an optimal a priori error estimate for stream variable under piecewise H 1 -norm is obtained. It is shown that all results are uniform with respect to the mesh size, and the location of the interface relative to the mesh. Finally, numerical experiments are carried out to demonstrate our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, a highly efficient space–time operator splitting finite element method is presented to solve the two- and three-dimensional Allen–Cahn equations. The main advantage of the proposed method is that it reduces the high storage requirements and complexity of the high-dimensional computation by splitting the high-dimensional problem into a series of one-dimensional subproblems. The proposed method is space–time second-order and can be performed in parallel. Moreover, a bound preserving least-distance modification technique is developed to force the discrete maximum bound principle in solving each one-dimensional subproblem. Finally, numerical simulations including the two- and multi-phase separations, mean curvature flows and dendritic crystal growth in two and three dimensions are provided to demonstrate the validity and accuracy of the proposed method. • A parallel operator splitting method is presented to solve Allen–Cahn equations. • A least-distance modification technique is developed to force the discrete MBP. • Simulations of phase separation and dendritic crystal growth are performed. [ABSTRACT FROM AUTHOR]

Wavelets provide sparse representations of integral and differential operators through a multiresolution approximation. This property is well-suited for adaptive implementation of numerical techniques, which is a common topic among researchers nowadays. The current state of the wavelet theory also allows flexible construction of transforms customized to certain applications, which is an important feature considering that the generalization of wavelet techniques to a large class of real-world numerical simulations is still limited. In this paper, it is proposed an adaptive wavelet finite element method for achieving operator-orthogonality between approximation spaces, a property which eliminates coupling among levels of resolution – a common shortcoming of adaptive methods. The resulting method is easily extended to any given finite element application and requires less computation power when compared to other adaptive techniques. These features are showcased by comparing the proposed technique to the popular h -refinement method using electromagnetic device simulations involving Dirichlet and Neumann boundary conditions, inhomogeneous and non-linear media, and irregular meshes. • This paper presents a general technique that allows the adaptive implementation of operator-orthogonal wavelets in any given simulation. • Simulations of electromagnetic devices are presented in order to prove the effectiveness of the proposed method. • The proposed method has shown significant improvements in runtime and specially memory usage – up to fifteen times less memory required – over the traditional h -refinement technique. [ABSTRACT FROM AUTHOR]

This paper aims to develop a semi-Lagrangian Bernstein–Bézier high-order finite element method for solving the two-dimensional nonlinear coupled Burgers' equations at high Reynolds numbers. The proposed method combines the semi-Lagrangian scheme for the time integration and the high-order Bernstein–Bézier functions for the space discretization in the finite element framework. Unstructured triangular Bernstein–Bézier patches are reconstructed in a simple and inherent manner over finite elements along the characteristic curves defined by the material derivative. A fourth-order Runge–Kutta scheme is used for the approximation of departure points along with a local L 2 -projection to compute the solution at the semi-Lagrangian stage. By using these techniques, the nonlinear problem is decoupled and two linear diffusion problems are solved separately for each velocity component. An implicit time-steeping scheme is used and a preconditioned conjugate gradient solver is used for the resulting linear systems of algebraic equations. The proposed method is investigated through several numerical examples including convergence studies. It is found that the proposed method is stable, highly accurate and efficient in solving two-dimensional coupled Burgers' equations at high Reynolds numbers. [ABSTRACT FROM AUTHOR]

This paper describes the mathematical modeling of a parabolic trough collector where the heat transfer fluid has temperature dependent physical properties for which we have developed our own mathematical expressions. For a specific location and date, the direct solar radiation and other sun–earth parameters are computed on a Julian day basis. Also, we have used certain known and recent correlations for the heat transfer coefficients appearing in this model. The mathematical model is governed by three nonlinear partial differential equations of parabolic type with initial/boundary value conditions. We describe a numerical algorithm in order to obtain the approximate solution of this model. Then, we carry out some numerical simulations which yield valuable information for the performance and efficiency of a solar PTC plant at the specific location. [ABSTRACT FROM AUTHOR]

In this paper, we obtain the expressions of epidemiological parameters using a class of feedforward neural networks (FNNs) based on the data collected from Chinese Center for Disease Control and Prevention (CCDCP), and establish a stochastic age-structured HIV/AIDS model with parameter perturbation. In order to ensure the mathematical and epidemiological rationality of the model, we discuss the existence, uniqueness, and boundedness of its positive solution. Due to the difficulty of solving the true solution for the stochastic age-structured HIV/AIDS model, we propose a full-discrete scheme using the Galerkin finite element method in age discretization and the Euler's scheme in time discretization. The error estimation between the numerical solution and the true solution is established. The results are justified by computer simulations. [ABSTRACT FROM AUTHOR]

This paper presents a stabilization technique for the finite element modelling of contact-impact problems of elastic bars via a bi-penalty method for enforcing contact constraints while employing an explicit predictor–corrector time integration algorithms. The present proposed method combines three salient features in carrying out explicit transient analysis of contact-impact problems: the addition of a penalty term associated with a kinetic energy expression of gap constraints, in addition to the conventional internal energy penalty term of the gap constraints; an explicit integration method that alleviates spurious oscillations; and, a judicious selection of two penalty parameters such that the stable time steps of the resulting explicit method is least compromised. Numerical experiments have been carried out with three explicit methods: the standard central difference method, the stabilized predictor–corrector method (Wu, 2003 [50]) and a method for mitigating spurious oscillations (Park et al., 2012 [44]) as applied to simulate one-dimensional contact-impact problems of the Signorini problem and the impact of two elastic bars. Results indicate that the proposed method can maintain the contact-free stability limit of the central difference and yield improved accuracy compared with existing bi-penalty methods. [ABSTRACT FROM AUTHOR]

Advanced numerical analysis of structures forces the massive implementation of numerical algorithms, relying on certain convergence of sequences of discrete finite element and similar approximations in spaces of integrable functions, conceding non-physical discontinuities and imperfectness to all numerical results. This is demonstrated on a model example of a thick elastic beam on the 2-parametric Pasternak subsoil. A potential remedy, discussed in this paper, comes from the implementation of available knowledge of analytical solutions in standard variational formulations. The resulting sparse system of linear algebraic equations can be derived without any numerical differentiation; the numerical quadrature is limited to some integrals of explicitly known (MAPLE-generated) functions. Potential generalizations, including selected nonlinear problems, are sketched. [ABSTRACT FROM AUTHOR]

The paper deals with the numerical realization of the 3D Stokes flow subject to threshold slip boundary conditions. The weak velocity–pressure formulation leads to an inequality type problem that is approximated by a mixed finite element method. The resulting algebraic system is non-smooth. Besides the pressure, three additional Lagrange multipliers are introduced: the discrete normal stress releasing the impermeability condition and two discrete shear stresses regularizing the non-smooth slip term. Eliminating the discrete velocity component we obtain the minimization problem for the smooth functional, expressed in terms of the pressure, the normal, and the shear stresses. This problem is solved either by a path following variant of the interior point method or by the semi-smooth Newton method. Numerical scalability is illustrated by computational experiments. [ABSTRACT FROM AUTHOR]

In this paper, wave propagation in a two-layer composite strip is investigated analytically and numerically. The strip is loaded by a very short transverse stress pulse. Three cases of the strip problem are assumed: (i) isotropic aluminum Al strip and two-layer strips made of (ii) Al and the ceramics Al 2 O 3 and (iii) the ceramics Al 2 O 3 and Al. The analytical method is based on Laplace and Fourier transform. The in-house finite element algorithm and thermodynamic consistent finite volume scheme are employed for computations, while the explicit time stepping procedure is used in both cases. The comparison of analytical and numerical results determines the degree of the accuracy of calculations, which is important for simulation of complex wave propagation problems in general heterogeneous media. [ABSTRACT FROM AUTHOR]

In this paper, we develop the constraint energy minimization generalized multiscale finite element method (CEM-GMsFEM) in mixed formulation applied to parabolic equations with heterogeneous diffusion coefficients. The construction of the method is based on two multiscale spaces: pressure multiscale space and velocity multiscale space. The pressure space is constructed via a set of well-designed local spectral problems, which can be solved independently. Based on the computed pressure multiscale space, we will construct the velocity multiscale space by applying constrained energy minimization. The convergence of the proposed method is proved. In particular, we prove that the convergence of the method depends only on the coarse grid size, and is independent of the heterogeneities and contrast of the diffusion coefficient. Four typical types of permeability fields are exploited in the numerical simulations, and the results indicate that our proposed method works well and gives efficient and accurate numerical solutions. [ABSTRACT FROM AUTHOR]

This paper attempts to merge the concept of hierarchical finite element analysis (FEA) into isogeometric analysis (IGA). The proposed methodology replaces the traditional grid refinement of IGA with custom enrichment functions. The enrichment functions are properly designed B-Spline wavelets tailored to eliminate scale-coupling terms in the stiffness matrix. In this way, the refined solution is synthesized from contributions of smaller independent problems. The proposed approach has two obvious benefits: (1) the calculations performed at each resolution are not discarded when proceeding to a finer one, and (2) it has less computational requirements since the solution is divided into smaller systems. Numerical results on an elasticity problem demonstrate superior performance and accuracy compared to traditional FEA and IGA schemes. [ABSTRACT FROM AUTHOR]

This paper presents a modified graded mesh for singularly perturbed reaction–diffusion problems. The mesh we offer is generated recursively using Newton's algorithm and some implicitly defined function. The problem is solved numerically using the finite element method based on polynomials of degree p ≥ 1. We prove parameter uniform convergence of optimal order in ϵ -weighted energy norm. Test examples are taken, and we present a rigorous comparative analysis with other adaptive meshes. Moreover, we compare the proposed method with others found in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we propose a new phase field model involving the p -Laplacian operator with 1 < p ≤ 2. This newly developed model can be used to simulate the sub-diffuse interface phenomena in the course of phase transitions during microstructure evolution. We also propose an energy-stable finite element approximation for the proposed nonlinear parabolic problem and rigorously prove that the finite element approximation is unconditionally energy stable. Moreover, we give the optimal error estimate and convergence rate with respect to the mesh sizes. Numerical examples are carried out to demonstrate the stability and accuracy of the proposed scheme, which is in accordance with the theoretical ones. Numerical results also indicate that the evolution of free energy is in consistency with the discrete dissipation theory. Moreover, the energy initially decreases sharply when the parameter p becomes larger. Meanwhile, we can observe that the parameter p affects the spatial patterns during phase transition. More precisely, the phase field value will be changed more slowly as well when the parameter p becomes smaller. [ABSTRACT FROM AUTHOR]

In this paper, two efficient two-grid algorithms with L 1 scheme are presented for solving two-dimensional nonlinear time fractional diffusion equations. The classical L 1 scheme is considered in the time direction, and the two-grid FE method is used to approximate spatial direction. To linearize the discrete equations, the Newton iteration approach and correction technique are applied. The two-grid algorithms reduce the solution of the nonlinear fractional problem on a fine grid to one linear equation on the same fine grid and an original nonlinear problem on a much coarser grid. As a result, our algorithms save total computational cost. Theoretical analysis shows that the two-grid algorithms maintain asymptotically optimal accuracy. Moreover, the numerical experiment presented further confirms the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, an efficient method seeking the numerical solution of a time-fractional convection equation whose solution is not smooth at the starting time is presented. The Caputo time-fractional derivative of order in (0 , 1) is discretized by the L1 finite difference method using non-uniform meshes; and, for the spatial derivative the discontinuous Galerkin (DG) finite element method is used. The stability and convergence of the method are analyzed for two-dimensional domains, using Cartesian and a particular class of unstructured grids. At last, several numerical examples are carried out which support the theoretical analysis. [ABSTRACT FROM AUTHOR]

In this paper, we use the finite element method to construct a new normalized basis of a univariate C 2 cubic spline space endowed with a specific subdivision of a real interval. Based on the polar forms, we introduce a new representation of the Hermite interpolant of any C 2 piecewise polynomial defined over this subdivision and we construct several superconvergent discrete quasi-interpolants which have an optimal approximation order. This approach is simple and provides an interesting approximation. Numerical results are given to illustrate the theoretical ones. [ABSTRACT FROM AUTHOR]

This paper deals with pricing of European and American options, when the underlying asset price follows Heston model, via the interior penalty discontinuous Galerkin finite element method (dGFEM). The advantages of dGFEM space discretization with Rannacher smoothing as time integrator with nonsmooth initial and boundary conditions are illustrated for European vanilla options, digital call and American put options. The convection dominated Heston model for vanishing volatility is efficiently solved utilizing the adaptive dGFEM. For fast solution of the linear complementary problem of the American options, a projected successive over relaxation (PSOR) method is developed with the norm preconditioned dGFEM. We show the efficiency and accuracy of dGFEM for option pricing by conducting comparison analysis with other methods and numerical experiments. [ABSTRACT FROM AUTHOR]

A static and dynamic large deformation analysis of a tapered beam subjected to concentrated and distributed loads is presented in this paper by using a direct integration technique. The bending stiffness of the beam is coordinate dependent. The nonlinear differential equation is numerically solved using an iterative technique without an algebraic equation solver, thus the computational effort can be reduced. A concentrated mass fixed at the free end and suddenly released is studied, and the time-dependent displacements are presented. Comparison has been made with solutions obtained using Finite Element Analysis and excellent agreement is achieved. [ABSTRACT FROM AUTHOR]

In this paper, the Galerkin method, based on cubic B-spline function as the shape and weight functions is applied for the numerical solution of the one-dimensional coupled nonlinear Schrödinger equation. Numerical experiments involving single solitary wave, collision of two solitary waves and collision of three solitary waves are conducted. The obtained numerical results of the proposed scheme are compared with the analytical results and previously published numerical results. Two conserved quantities I 1 and I 2 are calculated for collision of two solitary waves and interaction of three solitary waves. The scheme provides accurate results which are in good agreement when compared to other numerical schemes. The order of convergence of the scheme is calculated. Moreover, the use of cubic B-spline Galerkin method produces smooth solutions without numerical smearing. [ABSTRACT FROM AUTHOR]

In order to avoid mechanical resonance, a vibrating structure needs to be designed such that its working frequency interval is sufficiently far from its natural frequencies. Natural frequencies of a mechanical system are obtained from free vibration analysis commonly done by the finite element method. One important step in the analysis is matrix tridiagonalization commonly performed by the block Lanczos method. However, the classical block Lanczos method suffers numerical instability due to the loss of matrix orthogonality. In this paper, we demonstrate the implementation of the block Lanczos method with an orthogonality fixing scheme for 3D free vibration problems. The solution accuracy and computational time of this method are compared with those of the classical block Lanczos method and the Householder method. The results show that the block Lanczos method with the orthogonality fixing scheme employed in this work can effectively avoid the numerical instability due to the loss of matrix orthogonality. Furthermore, the block Lanczos method with the orthogonality fixing scheme provides solution accuracy as good as the Householder method while using significantly less computational time. [ABSTRACT FROM AUTHOR]

This paper studies an optimal control problem of a susceptible–infected–recovered (SIR) reaction–diffusion model to derive an efficient vaccination strategy for influenza outbreaks. The control problem reflects realistic restrictions associated with limited total vaccination coverage and the maximum daily vaccine administration using state variable inequality constraints. We prove the existence of the optimal control solution and also investigate an optimality system by introducing a penalty function to deal with the constrained optimal control problem. A gradient-based algorithm is discussed to solve the optimality system. The spatial SIR model is solved by using the finite difference method (FDM) in time and the finite element method (FEM) in space. The results of numerical simulations show that the optimal vaccine strategy varies regionally according to the spreading rate of the disease. [ABSTRACT FROM AUTHOR]

In this paper, efficient methods for numerical solutions of Caputo-type nonlinear conservation laws are established and studied, where the time fractional derivative with order in (0 , 1) is discretized by the finite difference method and the spatial derivative by the discontinuous Galerkin finite element method. The derived numerical schemes for one and two space dimensions are shown to be stable and convergent. Numerical experiments are provided to support these conclusions. [ABSTRACT FROM AUTHOR]

In this paper, we present a two-grid partially penalized immersed finite element (IFE) scheme for the approximation of semi-linear elliptic interface problems. Extra stabilization terms are introduced at interface edges for penalizing the discontinuity in IFE functions. Optimal error estimates in both H 1 and L p norms are obtained for IFE discretizations. To linearize the IFE equations, two-grid algorithm based on some Newton iteration approach is applied. It is shown that the coarse grid can be much coarser than the fine grid and achieve asymptotically optimal approximation as long as the mesh sizes satisfy H = O (h 1 ∕ 4). As a result, solving such a large class of non-linear equation will not be much more difficult than solving one single linearized equation. [ABSTRACT FROM AUTHOR]

Large number of measurement techniques involves the scanning of a probe, while a physical quantity is measured. Data acquisition therefore requires high acquisition rates, together with an accurate synchronization between the probe control and the measurement. LabVIEW-based graphical programming language provides the flexibility of integration of data acquisition software/hardware for automated test and measurement applications using National Instruments (NI) high-speed data acquisition boards. This paper describes a new LabVIEW-Based automatic test system that can be used for eddy current testing (ECT) tests and measurements. The system deals with a pancake type coil, placed above a flat conductive plate with a crack. In order to achieve the crack detection, the probe scans the target and signals are collected by a NI data acquisition board. The software of the system has been developed using LabVIEW in order to realize the real time data acquisition of coil's impedance. To verify the experimental results, simulations were performed using 3D finite element method. The system designed can be used for both education and specific research purposes. [ABSTRACT FROM AUTHOR]

Abstract: The paper deals with the dynamic modeling of mechatronic devices, which usually need detailed modeling to be described and to take into account the physical properties of the VHDL-AMS, which is a powerful unified modeling language for mixed system, allows to describe a large range of physical systems, for their dynamic simulation. It allows to describe models of physical components and then to connect them to obtain the model of a system. However, this language cannot support the description of some physical phenomena, such local ones, defined by numerical methods (e.g.: finite element method, special numerical integrals). When an aspect of a model cannot be described in VHDL-AMS, the paper proposes to use software components. So, the aim of the paper is to propose a generic way to extend the computation capability of VHDL-AMS, by coupling the models described in VHDL-AMS with external ones specified as software components (where VHDL-AMS fails). The approach has been applied on several applications, among them the time simulation of an electrical plunger. [Copyright &y& Elsevier]

Abstract: Topology optimization methods are aimed to produce optimal design. These tools implement optimization algorithms that modify the distribution of some materials within a predefined design space without a priori ideas regarding the topology or the geometry of the best solution. In this paper, we study a specific tool that combines a genetic algorithm, a material distribution formalism based on Voronoi cells and a commercial FEM evaluation tool. In particular, this paper shows, through a simple but representative case study, that it is possible to improve the performance of the topology optimization tool during the local search phase, i.e. the geometrical and dimensional optimization phase for which the topology optimization methods are originally not well-suited. [Copyright &y& Elsevier]

Abstract: The paper deals with a model of the homogenized fluid saturated porous material which recently was obtained by the authors using the asymptotic analysis of the Biot type medium characterized by the double porosity. The homogenized macroscopic model is featured by the fading memory effects arising from the microflow in the dual porosity. We derive the steady state formulations and discuss several topics related to the numerical implementation of the model, namely the solution procedure of the discretized microscopic problems, evaluation of the homogenized coefficients and an approximation of the convolution integrals of the macroscopic model, so that the fading memory effects are computationally tractable. Numerical examples are presented to illustrate the approximation schemes discussed in the paper. All computations were performed using the in-house developed finite element code SfePy allowing the multiscale simulations. Besides various potential engineering applications, the present model is intended for simulations of compact bone poroelasticity. [Copyright &y& Elsevier]

Abstract The aim of this paper is to develop a new non-boundary fitted (NBF) Cartesian grid method to compute flows in a complex geometry such as that found in a human aorta. The algorithm is based on the volume of fluid (VOF) colour function in a regular Cartesian grid through the open source OpenFOAM platform. The solid was treated as a fluid with very high viscosity to theoretically reduce its deformability, and subsequently satisfy a no-slip condition at the boundary. We also introduced the artificial term of the colour function into an algebraic system of momentum equations, which had a significant impact on the non deformability of solid domain and originality of this study. The developed solver, NBF-VOF, is then thoroughly validated using a variety of numerical and experimental results available in the literature Finally, we demonstrated the applicability of our code on flows in a complex geometry based on patient-specific image data robustly without the need for complex meshing generation procedures that conventional boundary fitted (BF) methods do. Highlights • Use VOF colour function to distinguish between solid and fluid. • Non movable solid is treated with high viscous fluid and modification to the systems of linear algebraic equations. • The developed method was validated through CFD canonical test on varied Reynolds Number. • The developed method was applied to real human aorta. • Free open Source software OpenFOAM is used as solver platform. [ABSTRACT FROM AUTHOR]

Abstract Park models, also known as dq-models, are a simple and very efficient way to obtain a faithful image of the behaviour of a machine in transient and steady-state operating conditions to deal with control issues. Their precision is based on the use of saturated direct and quadrature reactances that truly represent the dynamics and the magnitude of the electrical signals at nominal operating point. However, these constant parameters are not sufficient to accurately take into account the saliency of the machine and, therefore, to precisely simulate the harmonic contents of these signals: the latter point can be very problematic for diagnostic purposes. This paper uses simulation results and experimental measurements on a Leroy-Somer 27kVA alternator to clearly show the lack of coherence between the simulated spectra from a dq-model and the experimental data retrieved from a real power alternator. For that reason, a new methodology of machine modelling consisting of a co-simulation process between Flux2D and Matlab software is presented. The use of finite elements enables a fine identification of the inductances of the system. These take into account the geometrical specificities of both the exciter and the main generator and are then used in an analytical model developed in the Matlab environment. Spectral comparisons with experimental data demonstrate the improved accuracy of this new modelling method compared to dq-models in terms of harmonic contents. [ABSTRACT FROM AUTHOR]

Abstract: We describe a new method for generating meshes that minimize the gradient of a discretization error. The key element of this method is construction of a tensor metric from edge-based error estimates. In our papers we applied this metric for generating meshes that minimize the gradient of P 1-interpolation error and proved that for a mesh with N triangles, the L 2-norm of gradient of the interpolation error is proportional to N −1/2. In the present paper we recover the tensor metric using hierarchical a posteriori error estimates. Optimal reduction of the discretization error on a sequence of adaptive meshes will be illustrated numerically for boundary value problems ranging from a linear isotropic diffusion equation to a nonlinear transonic potential equation. [Copyright &y& Elsevier]

In this paper, we introduce a variational integrator for higher order finite element models. The solution quality of finite element simulation strongly depends on the approximation in time and space. A variational integrator is robust due to conservation of both the balance of total linear momentum and the balance of total angular momentum. It also binds the error in the balance of total energy. We extend the variational integrator to an integrator that conserves the balance of total energy as well. However, the approximation in time cannot mend the approximation error in space. Therefore, we use a higher order approximation in space to obtain a better solution compared to the real live model that should be simulated. We show different numerical examples with Dirichlet and Neumann boundaries. For the Dirichlet boundaries we use the Lagrange multiplier method and for the Neumann boundaries we introduce the forces on the nodes by constant pressure. [ABSTRACT FROM AUTHOR]

Abstract: Detailed knowledge about the function and morphology of temporomandibular joint are necessary for clinical applications and for analyses of the function of temporomandibular joint prosthesis. Movements of temporomandibular joint are biomechanically sophisticated and are up-to-date not clear. Therefore, the aim of the paper is to give the suitable mathematical approach for analyses of temporomandibular joint. In the paper the analysis of the temporomandibular joint, loaded by traction and compression forces, is presented and shortly discussed. The obtained results, based on two- and three-dimensional mathematical models, represent an introductory work for further studies of biomechanical aspects of temporomandibular joint and of its artificial replacements. The models are based on the theory of contact problems in linear elasticity. For the numerical solutions of the investigated problems the FEM approaches are used and the used algorithm is based on an active-set method for quadratic programming. [Copyright &y& Elsevier]