Adaptive Higher-order Finite Element Research Articles

An adaptive order finite element method for poroelastic materials described through the Biot equations

AbstractPoroelastic materials are essential to engineers to improve the sound and vibration properties of a product. The most common mathematical model that describes the vibro‐acoustics of these materials is the Biot theory, which considers the fully coupled behavior of a homogenized solid phase, based on the structural skeleton, and a homogenized fluid phase, describing the inter‐penetrating fluid. This article focuses on an adaptive higher order finite element implementation to solve poroelastic models. Owing to the presence of several wave solutions, namely, two compressional wave types and one shear wave type, it is difficult to define clear meshing guidelines a priori for poroelastic material modeling and trial and error is required to ensure reliable results. Multiple scales may be present, and local resonance phenomena may significantly modify the resolution requirements in certain frequency bands. In order to remedy this, and to be able to deliver a (nearly) mesh‐independent solution, a physics based adaptivity mechanism is developed which determines the distribution of (directional) element orders at each frequency prior to the calculation, to reach a user‐defined target error. The resulting approach is verified for different element topologies and is further applied to relevant applications of poroelastic materials in vibro‐acoustics.

Solving elliptic eigenproblems with adaptive multimesh [formula omitted]-FEM

This paper proposes a novel adaptive higher-order finite element (hp-FEM) method for solving elliptic eigenvalue problems, where n eigenpairs are calculated simultaneously, but on individual higher-order finite element meshes. The meshes are automatically hp-refined independently of each other, with the goal to use an optimal mesh sequence for each eigenfunction. The method and the adaptive algorithm are described in detail. Numerical examples clearly demonstrate the superiority of the novel method over the standard approach where all eigenfunctions are approximated on the same finite element mesh.

An adaptive higher order finite element model and modal energy for the vibration of a traveling string

A nonlinear equation describing the transverse vibration of an axially traveling string with constant and time-varying length is obtained by developing a new finite element model described by quadratic shape functions. A novel nonlinear coordinate transform is introduced with regard to its nonlinear terms. Subsequently, a new hybrid Newmark-beta/time varying degree of freedom method, which can adjust the element number automatically according to the change of string length, is proposed to improve accuracy. The proposed method as well as normal numerical methods are compared with an analytical solution. Results show that the proposed method is in good agreement with the Newmark-beta method for the case of small variations in string length, whilst it is superior in accuracy to the latter in the case of large length variations. Complex mode theory is adopted firstly to obtain the modal components as well as subsequently the modal energy for a traveling string. A phenomenon is observed where the free vibration energy leaks from one mode to the others in a traveling string. The higher the speed of translation and the modal order, the more energy that is leaked into the modes close to the initially excited mode.

Fully adaptive higher-order finite element analysis of fast transient phenomena on overhead lines

An alternative fully adaptive finite element approach is used for modeling fast transient phenomena on overhead lines. The methodology is based on rewriting the hyperbolic transmission equation for voltage to two specific equations where one unknown is the voltage itself and another unknown is its partial derivative with respect to time. The system is then solved by the higher-order finite element method in space and by the backward differential formula in time. Computational tests carried out on several typical examples prove that the approach exhibits numerous advantages compared with finite difference time-domain method.

Errata to “Approximation classes for adaptive higher order finite element approximation”

Errata to “Approximation classes for adaptive higher order finite element approximation”

Hp-discontinuous Galerkin method based on local higher order reconstruction

We present a new adaptive higher-order finite element method (hp-FEM) for the solution of boundary value problems formulated in terms of partial differential equations (PDEs). The method does not use any information about the problem to be solved which makes it robust and equation-independent. It employs a higher-order reconstruction scheme over local element patches which makes it faster and easier to parallelize compared to hp-adaptive methods that are based on the solution of a reference problem on a globally hp-refined mesh. The method can be used for the solution of linear as well as nonlinear problems discretized by conforming or non-conforming finite element methods, and it can be combined with arbitrary a posteriori error estimators. The performance of the method is demonstrated by several examples carried out by the discontinuous Galerkin method.

Numerical Simulation of a Multi-Frequency Resistivity Logging-While-Drilling Tool Using a Highly Accurate and Adaptive Higher-Order Finite Element Method

A novel, highly efficient and accurate adaptive higher-order finite ele- ment 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. Numeri- cal 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.

High-speed rotation induction heating in thermal clamping technology

The clamping technology based on high-speed induction heating of the fixing part of the system is proposed and analyzed. The continuous mathematical model of the clamping process is given by three partial differential equations (PDEs) describing the distributions of electromagnetic field, temperature field and field of thermoelastic displacements. Their coefficients containing the physical parameters of the materials are nonlinear temperature-dependent functions. The system is solved numerically in the hard-coupled formulation using a fully adaptive higher-order finite element method developed by the hp-FEM group and implemented in the application Agros2D. The methodology is illustrated with a practical example of fixing the shank of a high-revolution drilling tool in the clamping head. The results are verified by experiments.

Induction heating of rotating nonmagnetic billet in magnetic field produced by high-parameter permanent magnets

An advanced way of induction heating of nonmagnetic billets is discussed and modeled. The billet rotates in a stationary magnetic field produced by unmoving high-parameter permanent magnets fixed on magnetic circuit of an appropriate shape. The mathematical model of the problem consisting of two coupled partial differential equations is solved numerically, in the monolithic formulation. Computations are carried out using our own code Agros2D based on a fully adaptive higher-order finite element method. The most important results are verified experimentally on our own laboratory device.

Evolutionary algorithm-based multi-criteria optimization of triboelectrostatic separator

A device for electrostatic separation of triboelectrically charged plastic particles is modeled and optimized. Electric field in the system is solved numerically by a fully adaptive higher-order finite element method. The movement of particles in the device is determined by an adaptive Runge–Kutta–Fehlberg algorithm. The shape optimization of the electrodes is carried out by a technique based on genetic algorithm NSGA-II and also on simulated annealing.

Arbitrary-level hanging nodes for adaptive [formula omitted]-FEM approximations in 3D

In this paper we discuss constrained approximation with arbitrary-level hanging nodes in adaptive higher-order finite element methods (hp-FEM) for three-dimensional problems. This technique enables using highly irregular meshes, and it greatly simplifies the design of adaptive algorithms as it prevents refinements from propagating recursively through the finite element mesh. The technique makes it possible to design efficient adaptive algorithms for purely hexahedral meshes. We present a detailed mathematical description of the method and illustrate it with numerical examples.

Hermes2D, a C++ library for rapid development of adaptive [formula omitted]-FEM and [formula omitted]-DG solvers

In this paper we describe Hermes2D, an open-source C++ library for the development and implementation of adaptive higher-order finite element and DG solvers for partial differential equations (PDE) and multiphysics coupled PDE problems. The library is suitable for applications ranging from simple linear PDE solvers to time-dependent solvers for nonlinear multiphysics coupled problems with dynamically changing meshes. We cover several typical application scenarios, and give a brief overview of methods and algorithms that the library provides. Numerical examples are provided.

Model of Induction Heating of Rotating Non-Magnetic Billets and its Experimental Verification

Induction heating of non-magnetic cylindrical billets is modeled. The billet rotates in static magnetic field generated by appropriately distributed permanent magnets. The mathematical model of the process consists of two second-order partial differential equations describing the distributions of magnetic and temperature fields. Its solution is solved numerically in the quasi-coupled formulation respecting all important non-linearities. The computations are carried out by the code Agros2-D, based on a fully adaptive higher-order finite element method. Some results are verified experimentally on an industrial prototype of the device. Presented are also other important results obtained by measurements.

Approximation classes for adaptive higher order finite element approximation

We provide an almost characterization of the approximation classes appearing when using adaptive finite elements of Lagrange type of any fixed polynomial degree. The characterization is stated in terms of Besov regularity, and requires the approximation within spaces with integrability indices below one. This article generalizes to higher order finite elements the results presented for linear finite elements by Binev et. al. [BDDP 2002].

Application of the hp-finite element method to modeling thermal fields of high voltage underground cables buried in multi-layer soil

In this paper, we investigate the application of the adaptive higher-order Finite Element Method (hp-FEM) to heat transfer problems in electrical engineering. The proposed method is developed based on the combination of the Delaunay mesh and higher-order interpolation functions. In which the Delaunay algorithm based on the distance function is used for creating the adaptive mesh in the whole solution domain and the higher-order polynomials (up to 9th order) are applied for increasing the accuracy of solution. To evaluate the applicability and effectiveness of this new approach, we applied the proposed method to solve a benchmark heat problem and to calculate the temperature distribution of some typical models of buried double- and single -circuit power cables in the homogenous and multi-layer soils, respectively.

Hp-FEM electromechanical transduction model of ionic polymer–metal composites

We propose a mathematically explicit model of ionic polymer–metal composite (IPMC) materials that consist of a coupled system of Poisson–Nernst–Planck–Euler equations, discretized by means of adaptive higher-order finite element methods (hp-FEM). We show that due to the transient character of the problem and different fields in terms of the locations of field gradients in the domain it is efficient to use adaptive algorithms that are capable of changing the mesh dynamically in time. We also show that due to large qualitative and quantitative differences between the four solution components–cation concentration, electric potential, x-directional displacement, and y-directional displacement–it is efficient to approximate them on different meshes using a novel adaptive multi-mesh hp-FEM. The study is accompanied with computations and comparisons of the adaptive multi-mesh hp-FEM with a number of adaptive FEM algorithms.

Separation of Plastic Particles in Electrostatic Field Produced by Electrodes of Optimized Shape

Shape optimization of electrodes for the device for electrostatic separation of triboelectrically charged plastic particles is carried out. The objective function maximizes the efficiency of separation consisting in the highest possible number of particles falling down to the prescribed bins. Electric field in the system is solved numerically, using the fully adaptive higher-order finite element method. The movement of particles in the device influenced by the Coulomb force is determined by means of an adaptive Runge-Kutta-Fehlberg method with a time varying time step. The shape optimization is carried out using a technique based on genetic algorithms. The methodology is illustrated by an example whose results are discussed.

On the shielding effectiveness calculation

Numerical analysis of the effectiveness of electromagnetic shielding is performed. Its first objective is to propose a new approach to the shielding effectiveness theory based on a high-frequency numerical model that uses integral characteristics in contrast with the standard model. The second goal is to find an appropriate method for the field computation. The mathematical model consists of partial differential equations describing the behaviour of an electromagnetic wave. Its numerical solution is performed by a fully adaptive higher-order finite element method developed by the hpfem.org group. The results are compared with data obtained by using commercial software COMSOL Multiphysics. The proposed method is illustrated by an example of shielding enclosures and some of the results are verified experimentally.

Modeling of loudspeaker using hp-adaptive methods

Numerical modeling of harmonic acoustic field produced by a common commercial loudspeaker is carried out. The goal is to obtain the directional characteristic of the device. So far, these characteristics could only be obtained by measurements, which was very complicated and also expensive. Another benefit of the model consists in obtaining a reference mathematical model for other acoustic devices, such as diffusors or resonators. The continuous mathematical model is derived from the physical laws and is considered for the ideal fluids with properties of a perfect continuum. The model of the loudspeaker in the axial symmetry (2D) is described by the Helmholtz differential equation for harmonic acoustic field and for the correct solution the appropriate boundary conditions and material properties are also defined. The task is solved numerically by a fully adaptive higher-order finite element method developed by the hp-FEM group. The convergence of results is discussed for the different adaptive techniques and the advantages of the use of \(hp\)-adaptivity are demonstrated. Selected results of the computations are then compared with experiments performed in an anechoic chamber at the University of West Bohemia and based on the standard CSN EN 60268-5.

Optimization of the system for induction heating of nonmagnetic cylindrical billets in rotating magnetic field produced by permanent magnets

The process of induction heating of cylindrical nonmagnetic billets is modeled and optimized. An unmovable billet is placed in rotating magnetic field generated by permanent magnets fixed in an external rotor driven by an asynchronous motor by means of a teeth gear. The coupled model of the problem consisting of two partial differential equations describing the distributions of magnetic and temperature fields in the system is solved by a fully adaptive higher-order finite element method. Computations are realized by own code Agros2D. The device is then optimized with respect to the total amount of heat generated in the billet; as the dimensions of the device are fixed, the only quantity that can be changed is the direction of magnetization of the individual permanent magnets. The optimization procedures represent a supplement to the code. Finally, the dynamic behavior of the optimized system is analyzed. The methodology is illustrated by a typical example whose results are discussed.