Adaptive Finite Research Articles

An Adaptive Finite Element/Finite Difference Domain Decomposition Method for Applications in Microwave Imaging

A new domain decomposition method for Maxwell’s equations in conductive media is presented. Using this method, reconstruction algorithms are developed for the determination of the dielectric permittivity function using time-dependent scattered data of an electric field. All reconstruction algorithms are based on an optimization approach to find the stationary point of the Lagrangian. Adaptive reconstruction algorithms and space-mesh refinement indicators are also presented. Our computational tests show the qualitative reconstruction of the dielectric permittivity function using an anatomically realistic breast phantom.

Convergence of Adaptive Weak Galerkin Finite Element Methods for Second Order Elliptic Problems

We consider a standard Adaptive weak Galerkin (AWG) finite element method for second order elliptic problems. We prove that the sum of the energy error and the scaled error estimator of AWG method, between two consecutive adaptive loops, is a contraction. At last, we present some numerical experiments to support the theoretical results.

Convergence and Quasi-Optimality of an Adaptive Continuous Interior Penalty Finite Element Method

Convergence and Quasi-Optimality of an Adaptive Continuous Interior Penalty Finite Element Method

Adaptive Finite Element‐Discrete Element Analysis for Microseismic Modelling of Hydraulic Fracture Propagation of Perforation in Horizontal Well considering Pre‐Existing Fractures

Hydrofracturing technology of perforated horizontal well has been widely used to stimulate the tight hydrocarbon reservoirs for gas production. To predict the hydraulic fracture propagation, the microseismicity can be used to infer hydraulic fractures state; by the effective numerical methods, microseismic events can be addressed from changes of the computed stresses. In numerical models, due to the challenges in accurately representing the complex structure of naturally fractured reservoir, the interaction between hydraulic and pre‐existing fractures has not yet been considered and handled satisfactorily. To overcome these challenges, the adaptive finite element‐discrete element method is used to refine mesh, effectively identify the fractures propagation, and investigate microseismic modelling. Numerical models are composed of hydraulic fractures, pre‐existing fractures, and microscale pores, and the seepage analysis based on the Darcy’s law is used to determine fluid flow; then moment tensors in microseismicity are computed based on the computed stresses. Unfractured and naturally fractured models are compared to assess the influences of pre‐existing fractures on hydrofracturing. The damaged and contact slip events were detected by the magnitudes, B‐values, Hudson source type plots, and focal spheres.

Efficient and Adaptive Orthogonal Finite Element Representation of the Geopotential

We unify and extend classical results from function approximation theory and consider their utility in astrodynamics. Least square approximation, using the classical Chebyshev polynomials as basis functions, is reviewed for discrete samples of the to-be-approximated function. We extend the orthogonal approximation ideas to n-dimensions in a novel way, through the use of array algebra and Kronecker operations. Approximation of test functions illustrates the resulting algorithms and provides insight into the errors of approximation, as well as the associated errors arising when the approximations are differentiated or integrated. Two sets of applications are considered that are challenges in astrodynamics. The first application addresses local approximation of high degree and order geopotential models, replacing the global spherical harmonic series by a family of locally precise orthogonal polynomial approximations for efficient computation. A method is introduced which adapts the approximation degree radially, compatible with the truth that the highest degree approximations (to ensure maximum acceleration error < 10−9 m s−2, globally) are required near the Earth’s surface, whereas lower degree approximations are required as radius increases. We show that a four order of magnitude speedup is feasible, with efficiency optimized using radial adaptation.

An Adaptive Hybrid Stress Transition Quadrilateral Finite Element Method for Linear Elasticity

In this paper, we discuss an adaptive hybrid stress finite element method on quadrilateral meshes for linear elasticity problems. To deal with hanging nodes arising in the adaptive mesh refinement, we propose new transition types of hybrid stress quadrilateral elements with 5 to 7 nodes. In particular, we derive a priori error estimation for the 5node transition hybrid stress element to show that it is free from Poisson-locking, in the

Adaptive Bilinear Element Finite Volume Methods for Second-Order Elliptic Problems on Nonmatching Grids

In this article, we propose and analyze two kinds of adaptive bilinear element finite volume methods for second-order elliptic problems on nonmatching grids. One of them chooses the piecewise bilinear finite element space as the trial function space, which is continuous on the matching part of a grid and is discontinuous on the nonmatching part of it. The other directly uses discontinuous piecewise bilinear element space for the trial function space. A priori estimations ensure the convergence and a posteriori estimations pave the way for adaptive methods. Several numerical experiments are presented to conform our theoretical results.

Adaptive Backstepping Finite Time Attitude Control of Reentry RLV with Input Constraint

This paper presents the finite time attitude tracking control problem of reusable launch vehicle (RLV) in reentry phase under input constraint, model uncertainty, and external disturbance. A control-oriented model of rotational dynamics is developed and used for controller design for the complex coupling of the translational and rotational dynamics. Firstly, fast terminal sliding mode control is incorporated into backstepping control to design controller considering model uncertainty and external disturbance. The “explosion of terms” problem inherent in backstepping control is eliminated by a robust second order filter. Secondly, the control problem in the presence of input constraint is further considered, and a constrained adaptive backstepping fast terminal sliding mode control scheme is developed. At the control design level, adaptive law is employed to estimate the unknown norm bound of lumped uncertainty with the reduction of computational burden. The Lyapunov-based stability analysis of the closed-loop system is carried out. The control performance is presented via the simulation of six-degree-of-freedom (6-DOF) model of RLV.

Adaptive Moving Mesh Finite Element Method for Space Fractional Advection Dispersion Equation

Space fractional advection dispersion equation includes non-local differential operators, which leads to calculating numerical integrals with weakly singular kernel on every subintervals. Oscillations are visible in computed solutions when fractional order tends to 1. Studies show that the stiffness matrix of time semi-discretization can be calculated directly by formulas established from a special variational formulation. Oscillations are eliminated by using adaptive moving mesh and De Boor algorithm, while the number of nodes remains unchanged.

Topology and Sizing Optimization of Trusses with Adaptive Ground Finite Elements Using Multiobjective PBIL

This paper presents an integrated design technique to carry out simultaneous topology and sizing optimization of a two-dimensional truss structure. An optimization problem is set to find a structural layout and elements’ cross-sectional areas of a 2D truss such that objective functions including mass, compliance, and buckling factor are optimized. Design constraints consist of stress, buckling, and compliance. The concept of an adaptive ground elements approach and the encoding/decoding process are detailed. The multiobjective version of population-based incremental learning (PBIL) is employed to solve the design problem. The results reveal that the proposed design strategy is efficient and can be an effective engineering design tool.

Unstructured Adaptive Triangular Mesh Generation Techniques and Finite Volume Schemes for the Air Bearing Problem in Hard Disk Drives

Unstructured adaptive triangular mesh generation techniques and vertex based finite volume schemes that suit slider air bearing simulation of hard disk drives are constructed and implemented. Different refinement and adaptation techniques are used to generate several levels of good quality mesh over sliders with complex rail shapes. At each level, either one geometrical or one physical property of the problem is captured. A group of implicit vertex based finite volume schemes is first constructed. The resulting simultaneous linear algebraic equations are solved iteratively by the Gauss-Seidel method. Unconditional stability of the scheme is achieved. In addition, we present a non-nested full approximation storage (FAS) multi-grid algorithm that can significantly speed up the convergence rate of the implicit finite volume schemes. The steady state flying attitude is obtained by a quasi-Newton iteration method. [S0742-4787(00)01804-X]

Quantitative Analysis of MR Brain Image Sequences by Adaptive Self-Organizing Finite Mixtures

This paper presents an adaptive structure self-organizing finite mixture network for quantification of magnetic resonance (MR) brain image sequences. We present justification for the use of standard finite normal mixture model for MR images and formulate image quantification as a distribution learning problem. The finite mixture network parameters are updated such that the relative entropy between the true and estimated distributions is minimized. The new learning scheme achieves flexible classifier boundaries by forming winner-takes-in probability splits of the data allowing the data to contribute simultaneously to multiple regions. Hence, the result is unbiased and satisfies the asymptotic optimality properties of maximum likelihood. To achieve a fully automatic quantification procedure that can adapt to different slices in the MR image sequence, we utilize an information theoretic criterion that we have introduced recently, the minimum conditional bias/variance (MCBV) criterion. MCBV allows us to determine the suitable number of mixture components to represent the characteristics of each image in the sequence. We present examples to show that the new method yields very efficient and accurate performance compared to expectation-maximization, K-means, and competitive learning procedures.

Parallel Domain Decomposition Solver for Adaptive hp Finite Element Methods

In this paper, the development and implementation of highly parallelizable domain decomposition solvers for adaptive hp finite element methods is discussed. Two-level orthogonalization is used to obtain a reduced system which is preconditioned by a coarse grid operator. The condition number of the preconditioned system, for Poisson problems in two space dimensions, is proved to be bounded by C(1 + Hp/h)2(1 + log p)2 and Cp(1 + Hp/h)2(1 + log p)2 for different choices of coarse grid operators, where H is the subdomain size, p is the maximum spectral order, h is the size of the smallest element in the subdomain, and C is a constant independent of the mesh parameters. The work here extends the work of Bramble et al. [Math Comp., 47 (1986), pp. 103--134] on the h-version and Babuska et al. [SIAM J. Numer. Anal., 29 (1991), pp. 624--661] on the p-version of the finite element method. A preliminary version of this solver was first announced by Oden, Patra, and Feng in [Domain Decomposition Solver for Adaptive hp Finite Elements, VII Conference on Domain Decomposition, State College, PA, October 1993]. Numerical experiments show fast convergence of the solver and good control of the condition number on a variety of discretizations.

An Adaptive Variational Finite Element Method for Computing Coupled Porous-Media Problems with Moving Fronts

SUMMARY An adaptive deforming-grid mixed-variable finite element method for solving two-dimensional porous-media transport problems governed by a system of coupled nonlinear partial differential evolution equations, is presented. The advantages of this methodology over standard fixed-grid formulations in improving solution accuracy and yielding better computational stability characteristics are discussed and demonstrated through solution of a representative set of problems. Accordingly, extremely accurate solutions were computed not only for the primary variables but also their fluxes, with a very small fixed number of deforming and moving finite elements, for a set of transient high Péclet number porous-media convection-diffusion (continuous moving front) problems and also for the front motion (discrete moving front) during resaturation by water of a porous rock.