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CONVErGENCE OF ADAPTIVE EDGE ELEMENT METhODS FOr ThE 3D EDDY CUrrENTS EQUATIONS

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Journal of Computational Mathematics 2009年第5期27卷 657-676页

作者： r.h.w.hoppe J.Schberl Institut fur Mathematik Universitat AugsburgD-86159AugsburgGermanyand Department of MathematicsUniversity of HoustonHoustonTX 77204-3008USA Dept.for Mathematics CCES(Center for Computational Engineering Science) RWTH AachenD-52074 AachenGermany

we consider an Adaptive Edge Finite Element Method （AEFEM） for the 3D eddy currents equations with variable coefficients using a residual-type a posteriori error estimator. Both the components of the estimator and certain oscillation terms, due to the occurrence of the variable coefficients, have to be controlled properly within the adaptive loop which is taken care of by appropriate bulk criteria. Convergence of the AEFEM in terms of reductions of the energy norm of the discretization error and of the oscillations is shown. Numerical results are given to illustrate the performance of the AEFEM.

关键词： Adaptive edge elements 3D eddy currents equations Convergence analysis,Error and oscillation reduction residual type a posteriori error estimates.

daptive hybridized Interior Penalty Discontinuous Galerkin Methods for h(curl)-Elliptic Problems

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Numerical Mathematics(Theory,Methods and Applications) 2011年第1期4卷 13-37页

作者： C.Carstensen r.h.w.hoppe N.Sharma T.warburton Department of Mathematics Humboldt Universität zu BerlinD-10099 BerlinGermany Department of Computer Science Engineering Yonsei UniversitySeoul 120-749Korea Department of Mathematics University of HoustonHouston TX 77204-3008USA Institute of Mathematics University of AugsburgD-86159 AugsburgGermany CAAM Rice UniversityHoustonTX 77005-1892USA.

we develop and analyze an adaptive hybridized Interior Penalty Discontinuous Galerkin(IPDG-h)method for h(curl)-elliptic boundary value problems in 2D or 3D arising from a semi-discretization of the eddy currents *** method can be derived from a mixed formulation of the given boundary value problem and involves a Lagrange multiplier that is an approximation of the tangential traces of the primal variable on the interfaces of the underlying triangulation of the computational *** is shown that the IPDG-h technique can be equivalently formulated and thus implemented as a mortar *** mesh adaptation is based on a residual-type a posteriori error estimator consisting of element and face *** a unified framework for adaptive finite element methods,we prove the reliability of the estimator up to a consistency *** performance of the adaptive symmetric IPDG-h method is documented by numerical results for representative test examples in 2D.

关键词： Adaptive hybridized Interior Penalty Discontinuous Galerkin method a posteriori error analysis h(curl)-elliptic boundary value problems semi-discrete eddy currents equations

A review of Unified A Posteriori Finite Element Error Control

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Numerical Mathematics(Theory,Methods and Applications) 2012年第4期5卷 509-558页

作者： C.Carstensen M.Eigel r.h.w.hoppe C.Löbhard Department of Mathematics Humboldt Universität zu BerlinD-10099 BerlinGermany Department of Computational Science and Engineering Yonsei UniversitySeoul 120-749Korea Department of Mathematics University of HoustonHouston TX 77204-3008USA. Institute of Mathematics University of AugsburgD-86159 AugsburgGermany

This paper aims at a general guideline to obtain a posteriori error estimates for the finite element error control in computational partial differential *** the abstract setting of mixed formulations,a generalised formulation of the corresponding residuals is proposed which then allows for the unified estimation of the respective dual ***,this can be done with an approach which is applicable in the same way to conforming,nonconforming and mixed ***,the unified approach is applied to various model *** particular,we consider the Laplace,Stokes,Navier-Lamé,and the semi-discrete eddy current equations.

关键词： A posteriori error analysis finite element method nonconforming finite element method mixed finite element method adaptive algorithm Poisson equation Laméequations Stokes equations Maxwell equations unified a posteriori error analysis discontinuous Galerkin residual estimator

OPTIMALITY OF LOCAL MULTILEVEL METhODS FOr ADAPTIVE NONCONFOrMING P1 FINITE ELEMENT METhODS

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Journal of Computational Mathematics 2013年第1期31卷 22-46页

作者： Xuejun Xu huangxin Chen r.h.w. hoppe LSEC ICMSEC Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100190 China School of Mathematical Sciences Xiamen University Xiamen 361005 China LSEC ICMSEC Academy of Mathematics and System Sciences Chinese Academy of Sciences Beijing 100190 China Department of Mathematics University of Houston Houston TX 77204-3008 USA Institute for Mathematicd University of Augsburg D-86159 Augsburg Germany

In this paper, a local multilevel product algorithm and its additive version are con- sidered for linear systems arising from adaptive nonconforming P1 finite element approx- imations of second order elliptic boundary value problems. The abstract Schwarz theory is applied to analyze the multilevel methods with Jaeobi or Gauss-Seidel smoothers per- formed on local nodes on coarse meshes and global nodes on the finest mesh. It is shown that the local multilevel methods are optimal, i.e., the convergence rate of the multilevel methods is independent of the mesh sizes and mesh levels. Numerical experiments are given to confirm the theoretical results.

关键词： Local multilevel methods Adaptive nonconforming P1 finite element methods,Convergence analysis Optimality.

ErrOr rEDUCTION IN ADAPTIVE FINITE ELEMENT APPrOXIMATIONS OF ELLIPTIC OBSTACLE PrOBLEMS

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Journal of Computational Mathematics 2009年第2期27卷 148-169页

作者： Dietrich Braess Carsten Carstensen ronald h.w.hoppe Faculty of Mathematics Ruhr University BochumD-44780 BochumGermany Institute of Mathematics Humboldt University of BerlinD-10099 BerlinGermany Department of Mathematics University of HoustonHoustonTX 77204-3008USA Institute of Mathematics University at AugsburgD-86159 AugsburgGermany

we consider an adaptive finite element method （AFEM） for obstacle problems associated with linear second order elliptic boundary value problems and prove a reduction in the energy norm of the discretization error which leads to r-linear convergence. This result is shown to hold up to a consistency error due to the extension of the discrete multipliers （point functionals） to h^-1 and a possible mismatch between the continuous and discrete coincidence and noncoincidence sets. The AFEM is based on a residual-type error estimator consisting of element and edge residuals. The a posteriori error analysis reveals that the significant difference to the unconstrained case lies in the fact that these residuals only have to be taken into account within the discrete noncoincidence set. The proof of the error reduction property uses the reliability and the discrete local efficiency of the estimator as well as a perturbed Galerkin orthogonality. Numerical results are given illustrating the performance of the AFEM.

关键词： Adaptive finite element methods Elliptic obstacle problems Convergence analysis

wEAK-DUALITY BASED ADAPTIVE FINITE ELEMENT METhODS FOr PDE-CONSTrAINED OPTIMIZATION wITh POINTwISE GrADIENT STATE-CONSTrAINTS

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Journal of Computational Mathematics 2012年第2期30卷 101-123页

作者： M. hintermuer Michael hinze ronald h.w. hoppe Department of Mathematics Humboldt-University of Berlin Germany Department of Mathematics University of Hamburg Germany Department of Mathematics University of Augsburg Germany and Department of MathematicsUniversity of Houston USA

Adaptive finite element methods for optimization problems for second order linear el- liptic partial differential equations subject to pointwise constraints on the l2-norm of the gradient of the state are considered. In a weak duality setting, i.e. without assuming a constraint qualification such as the existence of a Slater point, residual based a posteriori error estimators are derived. To overcome the lack in constraint qualification on the continuous level, the weak Fenchel dual is utilized. Several numerical tests illustrate the performance of the proposed error *** subject classification： 65N30, 90C46, 65N50, 49K20, 49N15, 65K10.

关键词： Adaptive finite element method A posteriori errors Dualization Low regu-larity Pointwise gradient constraints State constraints weak solutions.

Free vibration and transient response of double curved beams connected by intermediate straight beams

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Applied Mathematics and Mechanics(English Edition) 2025年第1期46卷 37-62页

作者： r.A.JAFArI-TALOOKOLAEI h.GhANDVAr E.JUMAEV S.KhATIr T.CUONG-LE School of Mechanical Engineering Babol Noshirvani University of Technology Department of Mechanical Engineering Faculty of Engineering New Uzbekistan University University of Business and Science Center for Engineering Application and Technology Solutions Ho Chi Minh City Open University

This paper investigates the free vibration and transient response of interconnected structures including double curved beams and intermediate straight beams,which are all constructed from functionally graded porous(FGP) materials. The strain potential and kinetic energies of each beam along with the work done by the external force are calculated. Additionally, a higher-order beam element is introduced to derive stiffness and mass matrices, along with the force vector. The curved and straight beams are discretized, and their assembled stiffness, mass matrices, and force vectors, are *** conditions at the joints are used to derive the total matrices of the entire structure. Subsequently, the natural frequencies and transient response of the system are determined. The accuracy of the mathematical model and the self-developed computer program is validated through the comparison of the obtained results with those of the existing literature and commercial software ANSYS, demonstrating excellent ***, a comprehensive study is conducted to investigate the effects of various parameters on the free vibration and transient response of the considered structure.

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MODELING,SIMULATION,AND OPTIMIZATION OF SUrFACE ACOUSTIC wAVE DrIVEN MICrOFLUIDIC BIOChIPS

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Journal of Computational Mathematics 2010年第2期28卷 149-169页

作者： harbir Antil roland Glowinski ronald h.w.hoppe Christopher Linsenmann Tsorng-whay Pan Achim wixforth Department of Mathematics University of Houston Institute of Mathematics University of Augsburg Institute of Physics University of Augsburg

we will be concerned with the mathematical modeling, numerical simulation, and shape optimization of micro fluidic biochips that are used for various biomedical applications. A particular feature is that the fluid flow in the fluidic network on top of the biochips is in- duced by surface acoustic waves generated by interdigital transducers. we are thus faced with a multiphysics problem that will be modeled by coupling the equations of piezoelectricity with the compressible Navier-Stokes equations. Moreover, the fluid flow exhibits a multiscale character that will be taken care of by a homogenization approach. we will discuss and analyze the mathematical models and deal with their numerical solution by space-time discretizations featuring appropriate finite element approximations with respect to hierarchies of simplicial triangulations of the underlying computational domains. Simulation results will be given for the propagation of the surface acoustic waves on top of the piezoelectric substrate and for the induced fluid flow in the microchannels of the fluidic network. The performance of the operational behavior of the biochips can be significantly improved by shape optimization. In particular, for such purposes we present a multilevel interior point method relying on a predictor-corrector strategy with an adaptive choice of the continuation steplength along the barrier path. As a specific example, we will consider the shape optimization of pressure driven capillary barriers between microchannels and reservoirs.

关键词： Microfluidic biochips Mathematical modeling Numerical simulation Shape optimization Multiphysics Multiscale problems.

A novel method for clustering cellular data to improve classification

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Neural regeneration research 2025年第9期20卷 2697-2705页

作者： Diek w.wheeler Giorgio A.Ascoli Center for Neural Informatics Structures&PlasticityKrasnow Institute for Advanced Study Bioengineering Department Volgenau School of Engineering George Mason University FairfaxVAUSA

Many fields,such as neuroscience,are experiencing the vast prolife ration of cellular data,underscoring the need fo r organizing and interpreting large datasets.A popular approach partitions data into manageable subsets via hierarchical clustering,but objective methods to determine the appropriate classification granularity are *** recently introduced a technique to systematically identify when to stop subdividing clusters based on the fundamental principle that cells must differ more between than within *** we present the corresponding protocol to classify cellular datasets by combining datadriven unsupervised hierarchical clustering with statistical *** general-purpose functions are applicable to any cellular dataset that can be organized as two-dimensional matrices of numerical values,including molecula r,physiological,and anatomical *** demonstrate the protocol using cellular data from the Janelia MouseLight project to chara cterize morphological aspects of neurons.

关键词： cellular data clustering dendrogram data classification Levene's one-tailed statistical test unsupervised hierarchical clustering