This paper presents a GPU-accelerated implementation of the Locally Optimal Block Preconditioned Conjugate Gradient (LOBPCG) method with an inexact nullspace filtering approach to find eigenvalues in electromagnetics analysis with higher-order FEM. The performance of the proposed approach is verified using the Kepler (Tesla K40c) graphics accelerator, and is compared to the performance of the implementation based on functions from the Intel MKL on the Intel Xeon (E5-2680 v3, 12 threads) central processing unit (CPU) executed in parallel mode. Compared to the CPU reference implementation based on the Intel MKL functions, the proposed GPU-based LOBPCG method with inexact nullspace filtering allowed us to achieve up to 2.9-fold acceleration.

A cheapest stable nonconforming finite element method is presented for solving the incompressible flow in a square cavity without smoothing the corner singularities. The stable cheapest nonconforming finite element pair based on P1×P0 on rectangularmeshes [29] is employed with a minimal modification of the discontinuous Dirichlet data on the top boundary, where is the finite element space of piecewise constant pressures with the globally one-dimensional checker-board pattern subspace eliminated. The proposed Stokes elements have the least number of degrees of freedom compared to those of known stable Stokes elements. Three accuracy indications for our elements are analyzed and numerically verified. Also, various numerous computational results obtained by using our proposed element show excellent accuracy.

In this paper, we propose a finite element method for the elasticity problems which have displacement discontinuity along the material interface using uniform grids. We modify the immersed finite element method introduced recently for the computation of interface problems having homogeneous jumps [20, 22]. Since the interface is allowed to cut through the element, we modify the standard Crouzeix-Raviart basis functions so that along the interface, the normal stress is continuous and the jump of the displacement vector is proportional to the normal stress. We construct the broken piecewise linear basis functions which are uniquely determined by these conditions. The unknowns are only associated with the edges of element, except the intersection points. Thus our scheme has fewer degrees of freedom than most of the XFEM type of methods in the existing literature [1,8,13]. Finally, we present numerical results which show optimal orders of convergence rates.

An optimal control problem is considered to find a stable surface traction, which minimizes the discrepancy between a given displacement field and its estimation. Firstly, the inverse elastic problem is constructed by variational inequalities, and a stable approximation of surface traction is obtained with Tikhonov regularization. Then a finite element discretization of the inverse elastic problem is analyzed. Moreover, the error estimation of the numerical solutions is deduced. Finally, a numerical algorithm is detailed and three examples in two-dimensional case illustrate the efficiency of the algorithm.

As an exploratory study for structural deformation and thermodynamic response induced by spacecraft reentry aerodynamic force and thermal environment, a finite element algorithm is presented on the basis of the classic Fourier heat conductive law to simulate the dynamic thermoelasticity coupling performance of the material. The Newmark method and Crank-Nicolson scheme are utilized to discretize the dynamic thermoelasticity equation and heat conductive equation in the time domain, respectively, and the unconditionally stable implicit algorithm is constructed. Four types of finite-element computing schemes are devised and discussed to solve the thermodynamic coupling equation, all of which are implemented and compared in the computational examples including the one-dimensional transient heat conduction in considering and not considering the vibration, the transient heat flow for the infinite cylinder, and the dynamic coupling thermoelasticity around re-entry flat plate from hypersonic aerothermodynamic environment. The computational results show that the transient responses of temperature and displacement field generate lag phenomenon in case of considering the deformation effect on temperature field. Propagation, rebounding, attenuation and stabilized phenomena of elastic wave are also observed by the finite-element calculation of thermodynamic coupling problem considering vibration and damping, and the oscillation of the temperature field is simultaneously induced. As a result, the computational method and its application research platform have been founded to solve the transient thermodynamic coupling response problem of the structure in strong aerodynamic heating and force environment. By comparing various coupling calculations, it is demonstrated that the present algorithm could give a correct and reliable description of transient thermodynamic responses of structure, the rationality of the sequentially coupling method in engineering calculation is discussed, and the bending deformation mechanism produced by the thermodynamic coupling response from windward and leeward sides of flying body is revealed, which lays the foundation in developing the numerical method to solve material internal temperature distribution, structural deformation, and thermal damage induced by spacecraft dynamic thermoelasticity coupling response under uncontrolled reentry aerothermodynamic condition.

This paper presents a relatively simple numerical method to investigate the flow and heat transfer of laminar power-law fluids over a semi-infinite plate in the presence of viscous dissipation and anisotropy radiation. On one hand, unlike most classical works, the effects of power-law viscosity on velocity and temperature fields are taken into account when both the dynamic viscosity and the thermal diffusivity vary as a power-law function. On the other hand, boundary layer equations are derived by Taylor expansion, and a mixed analytical/numerical method (a pseudosimilarity method) is proposed to effectively solve the boundary layer equations. This method has been justified by comparing its results with those of the original governing equations obtained by a finite element method. These results agree very well especially when the Reynolds number is large. We also observe that the robustness and accuracy of the algorithm are better when thermal boundary layer is thinner than velocity boundary layer.

In this paper, we study a new stabilized method based on the local pressure projection to solve the semi-linear elliptic equation. The proposed scheme combines nonconforming finite element pairs NCP1–P1 triangle element and two-level method, which has a number of attractive computational properties: parameter-free, avoiding higher-order derivatives or edge-based data structures, but have more favorable stability and less support sets. Stability analysis and error estimates have been done. Finally, numerical experiments to check estimates are presented.

In this paper, we introduce and study a new method for solving inverse source problems, through a working model that arises in bioluminescence tomography (BLT). In the BLT problem, one constructs quantitatively the bioluminescence source distribution inside a small animal from optical signals detected on the animal's body surface. The BLT problem possesses strong ill-posedness and often the Tikhonov regularization is used to obtain stable approximate solutions. In conventional Tikhonov regularization, it is crucial to choose a proper regularization parameter for trade off between the accuracy and stability of approximate solutions. The new method is based on a combination of the boundary condition and the boundary measurement in a parameter-dependent single complex Robin boundary condition, followed by the Tikhonov regularization. By properly adjusting the parameter in the Robin boundary condition, we achieve two important properties for our new method: first, the regularized solutions are uniformly stable with respect to the regularization parameter so that the regularization parameter can be chosen based solely on the consideration of the solution accuracy; second, the convergence order of the regularized solutions reaches one with respect to the noise level. Then, the finite element method is used to compute numerical solutions and a new finite element error estimate is derived for discrete solutions. These results improve related results found in the existing literature. Several numerical examples are provided to illustrate the theoretical results.

A numerical comparison of finite difference (FD) and finite element (FE) methods for a stochastic ordinary differential equation is made. The stochastic ordinary differential equation is turned into a set of ordinary differential equations by applying polynomial chaos, and the FD and FE methods are then implemented. The resulting numerical solutions are all non-negative. When orthogonal polynomials are used for either continuous or discrete processes, numerical experiments also show that the FE method is more accurate and efficient than the FD method.

We introduce a new multigrid method to study the lattice statics model arising from nanoindentation. A constrained Cauchy-Born elasticity model is used as the coarse-grid operator. This method accelerates the relaxation process and considerably reduces the computational cost. In particular, it saves quite a bit when dislocations nucleate and move, as demonstrated by the simulation results.

Taking vibration converter of intelligent damper for drill strings as the study object, this paper analyzes the influential factors of motion state of the ball and conducts an explicit dynamics simulation by establishing a mechanics model of vibration converter. The study basis is Newton’s laws of motion, d’alembert’s principle and hertz contact theory. and we use world coordinate system, rotating coordinate system and Frenet coordinate system to deduce kinematics equations of vibration converter. The ultimate result demonstrates that the axial velocity and maximum contact stress change with the increment of ball diameter and helix angle.It also proves the validity of our derived kinematics and mechanical models and provides a good consultant value for the design and theoretical arithmetic of vibration converter for intelligent damper of drill strings.

This paper proposes a variational binary level set method for shape and topology optimization of structural. First, a topology optimization problem is pre-sented based on the level set method and an algorithm based on binary level set method is proposed to solve such problem. Considering the difficulties of coordination between the various parameters and efficient implementation of the proposed method, we present a fast algorithm by reducing several parameters to only one parameter, which would substantially reduce the complexity of computation and make it easily and quickly to get the optimal solution. The algorithm we constructed does not need to re-initialize and can produce many new holes automatically. Furthermore, the fast algorithm allows us to avoid the update of Lagrange multiplier and easily deal with constraints, such as piecewise constant, volume and length of the interfaces. Finally, we show several optimum design examples to confirm the validity and efficiency of our method.

The shapes of vesicles formed by lipid bilayers with phase separation are governed by a bending energy with phase dependent material parameters together with a line energy associated with the phase interfaces. We present a numerical method to approximate solutions to the Euler-Lagrange equations featuring triangulated surfaces, isoparametric quadratic surface finite elements and the phase field approach for the phase separation. Furthermore, the method involves an iterative solution scheme that is based on a relaxation dynamics coupling a geometric evolution equation for the membrane surface with a surface Allen-Cahn equation. Remeshing and grid adaptivity are discussed, and in various simulations the influence of several physical parameters is investigated.

This paper summarizes suitable material models for creep and damage of concrete which are coupled with heat and moisture transfer. The fully coupled approach or the staggered coupling is assumed. Governing equations are spatially dis-cretized by the finite element method and the temporal discretization is done by the generalized trapezoidal method. Systems of non-linear algebraic equations are solved by the Newton method. Development of an efficient and extensible computer code based on the C++ programming language is described. Finally, successful analyses of two real engineering problems are described.

A novel, highly efficient and accurate adaptive higher-order finite element method (hp-FEM) is used to simulate a multi-frequency resistivity logging-while-drilling (LWD) tool response in a borehole environment. Presented in this study are the vector expression of Maxwell’s equations, three kinds of boundary conditions, stability weak formulation of Maxwell’s equations, and automatic hp-adaptivity strategy. The new hp-FEM can select optimal refinement and calculation strategies based on the practical formation model and error estimation. Numerical experiments show that the new hp-FEM has an exponential convergence rate in terms of relative error in a user-prescribed quantity of interest against the degrees of freedom, which provides more accurate results than those obtained using the adaptive h-FEM. The numerical results illustrate the high efficiency and accuracy of the method at a given LWD tool structure and parameters in different physical models, which further confirm the accuracy of the results using the Hermes library (http://hpfem.org/hermes) with a multi-frequency resistivity LWD tool response in a borehole environment.

We propose some new weighted averaging methods for gradient recovery, and present analytical and numerical investigation on the performance of these weighted averaging methods. It is shown analytically that the harmonic averaging yields a superconvergent gradient for any mesh in one-dimension and the rectangular mesh in two-dimension. Numerical results indicate that these new weighted averaging methods are better recovered gradient approaches than the simple averaging and geometry averaging methods under triangular mesh.

Flexible discretization techniques for the approximative solution of coupled wave propagation problems are investigated. In particular, the advantages of using non-matching grids are presented, when one subregion has to be resolved by a substantially finer grid than the other subregion. We present the non-matching grid technique for the case of a mechanical-acoustic coupled as well as for acoustic-acoustic coupled systems. For the first case, the problem formulation remains essentially the same as for the matching situation, while for the acoustic-acoustic coupling, the formulation is enhanced with Lagrange multipliers within the framework of Mortar Finite Element Methods. The applications will clearly demonstrate the superiority of the Mortar Finite Element Method over the standard Finite Element Method both concerning the flexibility for the mesh generation as well as the computational time.

In this paper, a new type of gradient recovery method based on vertex-edge-face interpolation is introduced and analyzed. This method gives a new way to recover gradient approximations and has the same simplicity, efficiency, and superconvergence properties as those of superconvergence patch recovery method and polynomial preserving recovery method. Here, we introduce the recovery technique and analyze its superconvergence properties. We also show a simple application in the a posteriori error estimates. Some numerical examples illustrate the effectiveness of this recovery method.

We propose a multigrid method to solve the molecular mechanics model (molecular dynamics at zero temperature). The Cauchy-Born elasticity model is employed as the coarse grid operator and the elastically deformed state as the initial guess of the molecular mechanics model. The efficiency of the algorithm is demonstrated by three examples with homogeneous deformation, namely, one dimensional chain under tensile deformation and aluminum under tension and shear deformations. The method exhibits linear-scaling computational complexity, and is insensitive to parameters arising from iterative solvers. In addition, we study two examples with inhomogeneous deformation: vacancy and nanoindentation of aluminum. The results are still satisfactory while the linear-scaling property is lost for the latter example.