Adaptive Higher-order Finite Element Method Research Articles

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.

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.

Finite-Element 2-D Model of Induction Heating of Rotating Billets in System of Permanent Magnets and its Experimental Verification

An alternative way of induction heating of nonmagnetic cylindrical billets is modeled. The billet rotates in static magnetic field generated by permanent magnets. The mathematical model of the process consisting of two second-order partial differential equations describing the distributions of magnetic and temperature fields is solved numerically in the quasi-coupled formulation. The computations are carried out by a fully adaptive higher order finite-element method that is implemented in own codes Agros2D and library Hermes. The most important results are verified experimentally on a prototype device built in our laboratory.

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.

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.

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.

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.

Numerical solution of coupled problems using code Agros2D

New code Agros2D for 2D numerical solution of coupled problems is presented. This code is based on the fully adaptive higher-order finite element method and works with library Hermes2D containing the most advanced numerical algorithms for the numerical processing of systems of second-order partial differential equations. It is characterized by several quite unique features such as work with hanging nodes of any level, multimesh technology (every physical field can be calculated on a different mesh generally varying in time) and a possibility of combining triangular, quadrilateral and curved elements. The power of the code is illustrated by three typical coupled problems.

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

AbstractA 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 automatichp-adaptivity strategy. The newhp-FEM can select optimal refinement and calculation strategies based on the practical formation model and error estimation. Numerical experiments show that the newhp-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 adaptiveh-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.

Induction heating of cylindrical nonmagnetic ingots by rotation in static magnetic field generated by permanent magnets

Induction heating of cylindrical nonmagnetic billets by their rotation in static magnetic field is modeled. The magnetic field is produced by a system of appropriately arranged permanent magnets. The numerical model is solved by our own full adaptive higher-order finite element method in a monolithic formulation, i.e., both magnetic and temperature fields are solved simultaneously, respecting their mutual interaction. All principal nonlinearities are included in the model (permeability of ferromagnetic parts of the system as well as temperature dependences of physical parameters of the heated metal). The methodology is illustrated by two examples whose results are discussed.

Numerical Model of Induction Shrink Fits in Monolithic Formulation

Induction heating-based assembly of axisymmetric shrink fits is modeled. The task represents a triply coupled evolutionary problem characterized by mutual interaction of electromagnetic field, temperature field, and field of thermoelastic displacements. Its numerical solution is performed by a fully adaptive higher-order finite element method in monolithic formulation, using a code developed by the authors. All nonlinearities of the system (magnetic permeability and temperature dependencies of all material parameters) are respected. The methodology is illustrated by a typical example whose results are discussed.

Adaptive <i>hp</i>-Finite Element Method for Electromagnetic Field Logging Problems

A novel, highly efficient and accurate adaptive higher-order finite element method (hp-FEM) is proposed for electromagnetic field problems. Presented in this paper 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. This method 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 methodology is freely available online in the form of a general public licensed C++ library Hermes (http://hpfem.org/hermes).

Space and Time Adaptive Two-Mesh hp-Finite Element Method for Transient Microwave Heating Problems

A novel, highly efficient and accurate space and time adaptive higher-order finite element method (hp-FEM) is proposed for evolutionary microwave heating problems. Since the electric field E and temperature field T are very different in nature, they are approximated on individual meshes that change dynamically in time, independently of each other. Although the approximations of E and T are defined on different meshes, the coupling is treated in a monolithic fashion using a complex-valued approximate temperature. Numerical experiments are presented that show the novel method is clearly superior to its natural competitors—the space and time adaptive (single mesh) hp-finite element method and the space and time adaptive two-mesh h-finite element method. In all cases, comparisons in both the number of degrees of freedom (discrete problem size) and CPU time are presented. The methodology is freely available on-line in the form of a general public licensed C++/Python library Hermes (http://hpfem.org/).

Adaptive [formula omitted]-FEM with dynamical meshes for transient heat and moisture transfer problems

We are concerned with the time-dependent multiphysics problem of heat and moisture transfer in the context of civil engineering applications. The problem is challenging due to its multiscale nature (temperature usually propagates orders of magnitude faster than moisture), different characters of the two fields (moisture exhibits boundary layers which are not present in the temperature field), extremely long integration times (30 years or more), and lack of viable error control mechanisms. In order to solve the problem efficiently, we employ a novel multimesh adaptive higher-order finite element method ( h p -FEM) based on dynamical meshes and adaptive time step control. We investigate the possibility to approximate the temperature and humidity fields on individual dynamical meshes equipped with mutually independent adaptivity mechanisms. Numerical examples related to a realistic nuclear reactor vessel simulation are presented.