Space-time adaptive $$hp$$ -FEM for problems with traveling sharp fronts

A hybrid method for modeling the creeping flow is proposed. In our method the so‐called marker‐in‐cell (MIC) and the Finite Element Method (FEM) algorithm are combined together to simulate the thermal creeping flow concerning heterogeneous medium deformation. In particular, the unknown parameters at the Euler mesh‐nodes are calculated using the FEM. The cell‐markers in each element carry the material composition and history variables during the flowing process. The momentum and continuity equation are solved in terms of the pressure‐stabilizing Petrov‐Galerkin method (PSPG) with the equal‐order interpolation of the velocity and pressure, and the energy equation is solved using the streamline upwind Petrov‐Galerkin method (SUPG). In the MIC algorithm, the bilinear interpolation corresponds to the interpolation function in the finite elements. The FEM and MIC algorithm are independent from each other. The data in these two processes communicate through the nodal points. In addition, the triangular element algorithm makes possible to solve the problems with irregular mesh‐grid in complex structures. Our computation program has been verified with the classical Rayleigh‐Benard convection problem. As an example, the descent of an irregular geometry block is calculated. The stability of numerical solution is also investigated.

Chapter 11 - Mixed finite element–computational fluid dynamics method for nonlinear fluid–solid interactions

Real-world groundwater modeling deals with heterogeneous and anisotropic aquifer systems. Numerical models, particularly the finite difference and finite element methods, are extensively used for modeling such a complex aquifer system. The finite difference method is an age-old straightforward approach, whereas the finite element method is more idealistic, relatively new, and has emerged as a popular approach. The unique advantages with the finite element method are (i) an irregular-shaped domain can be discretized into elements with a matching boundary, (ii) a node can be connected with the neighboring nodes in any direction. Thus, the finite element analysis allows water/solute to flow more freely toward the surrounding nodes along nodal links. On the other hand, the mathematical formulation of the finite element technique follows a set of rules step by step. Conventionally, this idealistic approach uses shape functions and integration techniques to develop element matrices, which, in turn, are assembled to form the global matrix. This conventional global matrix was found to be inconsistent with the basic partial differential equation representing the mass balance in a nodal domain as a control volume. As a result, the matrix solution, particularly for nonlinear problems (e.g., vadose zone flow, density dependent flow, and so forth) becomes unstable and computationally burdensome. To overcome the limitations of the conventional finite element method for real-world groundwater modeling, a new approach, called the direct-formulation finite element (DFFE) method, has been developed. In this approach, a two-dimensional model domain is discretized into triangular elements. Element matrices are formed directly based on flow principles and material properties of elements without using shape functions and integration process. The DFFE method is robust. The global matrix is consistent, and the solution is stable and efficient. The results obtained from this approach were found in a good agreement with that obtained from analytical and other widely used models.

Purpose – A variety of meshless methods have been developed in the last 20 years with an intention to solve practical engineering problems, but are limited to small academic problems due to associated high computational cost as compared to the standard finite element methods (FEM). The purpose of this paper is to develop an efficient and accurate algorithms based on meshless methods for the solution of problems involving both material and geometrical nonlinearities. Design/methodology/approach – A parallel two-dimensional linear elastic computer code is presented for a maximum entropy basis functions based meshless method. The two-dimensional algorithm is subsequently extended to three-dimensional adaptive nonlinear and three-dimensional parallel nonlinear adaptively coupled finite element, meshless method cases. The Prandtl-Reuss constitutive model is used to model elasto-plasticity and total Lagrangian formulations are used to model finite deformation. Furthermore, Zienkiewicz and Zhu and Chung and Belytschko error estimation procedure are used in the FE and meshless regions of the problem domain, respectively. The message passing interface library and open-source software packages, METIS and MUltifrontal Massively Parallel Solver are used for the high performance computation. Findings – Numerical examples are given to demonstrate the correct implementation and performance of the parallel algorithms. The agreement between the numerical and analytical results in the case of linear elastic example is excellent. For the nonlinear problems load-displacement curve are compared with the reference FEM and found in a very good agreement. As compared to the FEM, no volumetric locking was observed in the case of meshless method. Furthermore, it is shown that increasing the number of processors up to a given number improve the performance of parallel algorithms in term of simulation time, speedup and efficiency. Originality/value – Problems involving both material and geometrical nonlinearities are of practical importance in many engineering applications, e.g. geomechanics, metal forming and biomechanics. A family of parallel algorithms has been developed in this paper for these problems using adaptively coupled finite element, meshless method (based on maximum entropy basis functions) for distributed memory computer architectures.

In recent years Line Start Permanent Magnet (LSPM) motor has attracted a lot of attention thanks to many advantages, such as high efficiency, high power factor, and high power density. In this motor, at the start, the risk of irreversible demagnetization in Permanent Magnet (PM) is essential due to the large inrush current. So in this paper using Finite Element Method (FEM) the effective factors on demagnetization of this motor structure at various temperatures will be investigated. In addition, the sensitivity analysis will be carried out. Finally, the electromagnetic torque of this motor using a modified FEM algorithm will be calculated and compared to the conventional FEM algorithm.

Simulation of hyperelastic materials in real-time using deep learning.

BEM and FEM based numerical simulations for biomagnetic fluid flow

In optimal design of a permanent magnet (PM) motor, many design variables are required to consider some device properties. These variables are, for example, the shape of core and magnet, the tooth length, the number of turns, and the winding radius. Moreover, many restrictions must be considered in practical PM motor design. These restrictions are, for example, the slot space factor and the cogging torque. However, the optimization problem, which has many design variables by the finite element method (FEM), has not been reported. In this paper, the efficiency of PM motor under the considerations given above is optimized by using FEM and optimization algorithm. In this problem, an objective function has many local minima and it is difficult to calculate its gradient. For these reasons, the genetic algorithm (GA) and simulated annealing method (SA), which are stochastic methods, are used for optimization method, because of the possibility of global range search and because gradient calculation is not required. Adding to both optimization methods, in this paper, SA combined with GA is used for one of the optimization methods. It is found that the solutions optimized by these methods are reasonable from an engineering point of view. © 2001 Scripta Technica, Electr Eng Jpn, 138(3): 72–79, 2002

Vibration isolation design for periodically stiffened shells by the wave finite element method

The performance of electrical machines is largely dictated by the action of current and flux in the core length. The field in a cross-section obeys Poisson's equation and approximate solutions have been obtained by finite difference and element methods. The finite difference method requires a large number of nodes and is slow to converge as permeability is variable. The finite element method is more flexible being more readily fitted to iron-air boundaries and has better convergence. However, it is difficult to formulate a legitimate variational formulation for transient conditions in the presence of dissipation. Here, discrete equations are formed by applying Ampere's circuital law around each node. Careful choice of contour lines give a current distribution superior to that obtained with finite elements. Fast convergence is obtained and the method is applicable under transient conditions.

Summary form only given. Many materials in the nature or fabricated in laboratories exhibit nonlinear behavior where the permittivity and/or permeability is a function of the magnitude of the electric or magnetic field intensity. Such nonlinearity results in a nonlinear system of Maxwell's equations, which are more challenging to solve. Generally speaking, there are two kinds of methods commonly adopted to solve nonlinear problems. The first kind of methods solves the linearized versions of the original problems, while the second kind of methods deals with the nonlinear problems directly. Although more difficult to solve with a high efficiency, the second kind of methods is more desirable in order to capture the nonlinear phenomenon more accurately. Among various nonlinear solvers developed in the applied mathematics community, the Newton-Raphson method is widely used due to its second-order convergence, if started from an initial guess sufficiently close to the true solution. However, in a real application, there is no straightforward way to choose a good initial guess. As a result, the Newton-Raphson method could be unstable or divergent in solving complicated problems. Another difficulty of applying the Newton-Raphson method is that the Jacobian matrix needs to be updated and solved in every Newton iteration, which could be very time consuming for solving large problems. When the nonlinear electromagnetic problems are formulated and solved with the finite element method in time domain (J. M. Jin, The Finite Element Method in Electromagnetics, Second Edition. New York: Wiley, 2002), the numerical solution at a previous time step can be used as a natural choice for the initial guess of the Newton iteration at the successive time step. The convergent of the Newton-Raphson method can be achieved if the time step is small enough such that the solutions of two successive time steps are sufficiently close. In this presentation, we employ the time-domain finite element method (TDFEM) combined with the Newton-Raphson method and its variations to solve nonlinear electromagnetic problems. We first study the TDFEM formulation of nonlinear problems, and then show how it can be solved with Newton(-like) methods. We demonstrate and compare the performance of different nonlinear solvers including the basic Newton-Raphson method and its variations through a number of numerical examples.

The non-conformal domain decomposition algorithms of the hybrid finite element-boundary integral-multilevel fast multipole algorithm are presented for computing large electromagnetic scattering and radiation problems. The numerical performance of different transmission conditions between finite element method (FEM) subdomains, and FEM and boundary integral (BI) domains is studied in the paper. It is found that the Dirichlet transmission condition employed on the interface between FEM and BI domains has faster convergence than the Robin-type transmission condition. Furthermore, the cement-element-based dual-primal finite element tearing and interconnecting (FETI-DP) domain decomposition method has faster convergence than the Lagrange-multiplier-based FETI-DP method under the FE-BI system.

We design and analyze several Finite Element Methods (FEMs) applied to the Caffarelli-Silvestre extension that localizes the fractional powers of symmetric, coercive, linear elliptic operators in bounded domains with Dirichlet boundary conditions. We consider open, bounded, polytopal but not necessarily convex domains $\Omega \subset \mathbb{R}^d$ with $d=1,2$. For the solution to the extension problem, we establish analytic regularity with respect to the extended variable $y\in (0,\infty)$. We prove that the solution belongs to countably normed, power-exponentially weighted Bochner spaces of analytic functions with respect to $y$, taking values in corner-weighted Kondat'ev type Sobolev spaces in $\Omega$. In $\Omega\subset \mathbb{R}^d$, we discretize with continuous, piecewise linear, Lagrangian FEM ($P_1$-FEM) with mesh refinement near corners, and prove that first order convergence rate is attained for compatible data $f\in \mathbb{H}^{1-s}(\Omega)$. We also prove that tensorization of a $P_1$-FEM in $\Omega$ with a suitable $hp$-FEM in the extended variable achieves log-linear complexity with respect to $\mathcal{N}_\Omega$, the number of degrees of freedom in the domain $\Omega$. In addition, we propose a novel, sparse tensor product FEM based on a multilevel $P_1$-FEM in $\Omega$ and on a $P_1$-FEM on radical-geometric meshes in the extended variable. We prove that this approach also achieves log-linear complexity with respect to $\mathcal{N}_\Omega$. Finally, under the stronger assumption that the data is analytic in $\overline{\Omega}$, and without compatibility at $\partial \Omega$, we establish exponential rates of convergence of $hp$-FEM for spectral, fractional diffusion operators. We also report numerical experiments for model problems which confirm the theoretical results. We indicate several extensions and generalizations of the proposed methods.

TT-M FE method for a 2D nonlinear time distributed-order and space fractional diffusion equation

In this paper, we study the rectified linear unit (ReLU) artificial neural network (ANN) for grid adaptation in finite element method, which is used for solving differential equations (DEs) with initial/boundary condition. Compared with the classical adaptive finite element method (AFEM), ReLU ANN based on finite element method can keep the number of grid-points constant but change their relative location. Our numerical experiments show that approximate solutions obtained from the classical finite element method by ReLU ANN are more accurate than those obtained by AFEM.