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1.
Summary. Enhanced strain elements, frequently employed in practice, are known to improve the approximation of standard (non-enhanced) displacement-based elements in finite element computations. The first contribution in this work towards a complete theoretical explanation for this observation is a proof of robust convergence of enhanced element schemes: it is shown that such schemes are locking-free in the incompressible limit, in the sense that the error bound in the a priori estimate is independent of the relevant Lamé constant. The second contribution is a residual-based a posteriori error estimate; the L
2
norm of the stress error is estimated by a reliable and efficient estimator that can be computed from the residuals.
Mathematics Subject Classification (2000):65N30 相似文献
2.
C. Carstensen. 《Mathematics of Computation》2004,73(247):1153-1165
All first-order averaging or gradient-recovery operators for lowest-order finite element methods are shown to allow for an efficient a posteriori error estimation in an isotropic, elliptic model problem in a bounded Lipschitz domain in . Given a piecewise constant discrete flux (that is the gradient of a discrete displacement) as an approximation to the unknown exact flux (that is the gradient of the exact displacement), recent results verify efficiency and reliability of
in the sense that is a lower and upper bound of the flux error up to multiplicative constants and higher-order terms. The averaging space consists of piecewise polynomial and globally continuous finite element functions in components with carefully designed boundary conditions. The minimal value is frequently replaced by some averaging operator applied within a simple post-processing to . The result provides a reliable error bound with .
in the sense that is a lower and upper bound of the flux error up to multiplicative constants and higher-order terms. The averaging space consists of piecewise polynomial and globally continuous finite element functions in components with carefully designed boundary conditions. The minimal value is frequently replaced by some averaging operator applied within a simple post-processing to . The result provides a reliable error bound with .
This paper establishes and so equivalence of and . This implies efficiency of for a large class of patchwise averaging techniques which includes the ZZ-gradient-recovery technique. The bound established for tetrahedral finite elements appears striking in that the shape of the elements does not enter: The equivalence is robust with respect to anisotropic meshes. The main arguments in the proof are Ascoli's lemma, a strengthened Cauchy inequality, and elementary calculations with mass matrices.
3.
The optimal design problem for maximal torsion stiffness of an infinite bar of given geometry and unknown distribution of
two materials of prescribed amounts is one model example in topology optimisation. It eventually leads to a degenerate convex
minimisation problem. The numerical analysis is therefore delicate for possibly multiple primal variables u but unique derivatives σ : = DW(D
u). Even fine a posteriori error estimates still suffer from the reliability-efficiency gap. However, it motivates a simple
edge-based adaptive mesh-refining algorithm (AFEM) that is not a priori guaranteed to refine everywhere. Its convergence proof
is therefore based on energy estimates and some refined convexity control. Numerical experiments illustrate even nearly optimal
convergence rates of the proposed AFEM.
Supported by the DFG Research Center MATHEON “Mathematics for key technologies” in Berlin. 相似文献
4.
This paper discusses adaptive finite element methods for the solution of elliptic eigenvalue problems associated with partial differential operators. An adaptive method based on nodal-patch refinement leads to an asymptotic error reduction property for the computed sequence of simple eigenvalues and eigenfunctions. This justifies the use of the proven saturation property for a class of reliable and efficient hierarchical a posteriori error estimators. Numerical experiments confirm that the saturation property is present even for very coarse meshes for many examples; in other cases the smallness assumption on the initial mesh may be severe. 相似文献
5.
A posteriori error estimates for mixed FEM in elasticity 总被引:2,自引:0,他引:2
A residue based reliable and efficient error estimator is established for finite element solutions of mixed boundary value
problems in linear, planar elasticity. The proof of the reliability of the estimator is based on Helmholtz type decompositions
of the error in the stress variable and a duality argument for the error in the displacements. The efficiency follows from
inverse estimates. The constants in both estimates are independent of the Lamé constant , and so locking phenomena for are properly indicated. The analysis justifies a new adaptive algorithm for automatic mesh–refinement.
Received July 17, 1997 相似文献
6.
Ultrasonic heating of the skull 总被引:3,自引:0,他引:3
E L Carstensen S Z Child S Norton W Nyborg 《The Journal of the Acoustical Society of America》1990,87(3):1310-1317
Comparatively simple analysis shows that diagnostic ultrasound devices, in some cases, may approach output levels that can produce significant heating of tissues, particularly if the propagation path includes bone. Experimental tests of these predictions using rodents show that temperature increments of the order of 3 degrees C/W/cm2 are produced in skull bone with sharply focused fields at medically relevant frequencies. 相似文献
7.
8.
We present a pollution-free Petrov-Galerkin multiscale finite element method for the Helmholtz problem with large wave number κ. We use standard continuous Q1 finite elements at a coarse discretization scale H as trial functions. The test functions are the solutions of local problems at a finer scale h. The diameter of the support of the test functions behaves like mH for some oversampling parameter m. Provided m is of the order of log(κ) and h is sufficiently small, the resulting method is stable and quasi-optimal in the regime where H is proportional to κ−1. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
9.
Jarvis MW Daily JW Carstensen HH Dean AM Sharma S Dayton DC Robichaud DJ Nimlos MR 《The journal of physical chemistry. A》2011,115(4):428-438
The pyrolysis of 2-phenethyl phenyl ether (PPE, C(6)H(5)C(2)H(4)OC(6)H(5)) in a hyperthermal nozzle (300-1350 °C) was studied to determine the importance of concerted and homolytic unimolecular decomposition pathways. Short residence times (<100 μs) and low concentrations in this reactor allowed the direct detection of the initial reaction products from thermolysis. Reactants, radicals, and most products were detected with photoionization (10.5 eV) time-of-flight mass spectrometry (PIMS). Detection of phenoxy radical, cyclopentadienyl radical, benzyl radical, and benzene suggest the formation of product by the homolytic scission of the C(6)H(5)C(2)H(4)-OC(6)H(5) and C(6)H(5)CH(2)-CH(2)OC(6)H(5) bonds. The detection of phenol and styrene suggests decomposition by a concerted reaction mechanism. Phenyl ethyl ether (PEE, C(6)H(5)OC(2)H(5)) pyrolysis was also studied using PIMS and using cryogenic matrix-isolated infrared spectroscopy (matrix-IR). The results for PEE also indicate the presence of both homolytic bond breaking and concerted decomposition reactions. Quantum mechanical calculations using CBS-QB3 were conducted, and the results were used with transition state theory (TST) to estimate the rate constants for the different reaction pathways. The results are consistent with the experimental measurements and suggest that the concerted retro-ene and Maccoll reactions are dominant at low temperatures (below 1000 °C), whereas the contribution of the C(6)H(5)C(2)H(4)-OC(6)H(5) homolytic bond scission reaction increases at higher temperatures (above 1000 °C). 相似文献
10.
It is well known that solutions of the rational interpolation problem or Newton-Padé approximation problem can be represented with the help of continued fractions if certain normality assumptions are satisfied. By comparing two interpolating continued fractions, one obtains a recursive QD-type scheme for computing the required coefficients. In this paper a uniform approach is given for two different interpolating continued fractions of ascending and descending type, generalizing ideas of Rutishauser, Gragg, Claessens, and others. In the nonnormal case some of the interpolants are equal yielding so-called singular blocks. By appropriate “skips” in the Newton-Padé table modified interpolating continued fractions are derived which involve polynomials known from the Kronecker algorithm and from the Werner-Gutknecht algorithm as well as from the modification of the cross-rule proposed recently by the authors. A corresponding QD-type algorithm for the nonnormal Newton-Padé table is presented. Finally, the particular case of Padé approximation is discussed where—as in Cordellier's modified cross-rule—the given recurrence relations become simpler. 相似文献