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Soil slopes
Published in Rodrigo Salgado, The Engineering of Foundations, Slopes and Retaining Structures, 2022
In lower and upper bound analyses, it is essential to maximize the lower bound and minimize the upper bound solutions. To achieve this goal, a mathematical optimization technique (either linear or nonlinear programming) is required. The first work of this kind was done by Lysmer (1970), who found lower bound solutions to plane-strain soil mechanics problems using three-noded triangular elements and linear programming. The first formulation of the upper bound limit analysis using constant-strain triangular finite elements and linear programming was proposed by Anderheggen and Knöpfel (1972) for plate problems. Since Lysmer’s and Anderheggen and Knöpfel’s pioneering work, much work has been done on FELA with the aim of (1) reducing the computation time (Sloan 1988a, b, 1989), (2) simulating the yield criterion more realistically using nonlinear programming (Casciaro and Cascini 1982; Munro 1982; Jiang and Magnan 1997; Lyamin 1999), and (3) reducing the restrictions imposed on the number of possible velocity discontinuities in the optimization process (Sloan and Kleeman 1995). Kim et al. (2002) developed the complete set of techniques for performing limit analysis of slope stability problems with soil profile complexities of any kind, the presence of water, and the existence of boundary loadings.
Digital Video Coding Standards—MPEG-1/2 Video
Published in Yun-Qing Shi, Huifang Sun, Image and Video Compression for Multimedia Engineering, 2019
This section addresses the problem of determining the optimal MPEG [mpeg2] coding strategy in terms of the selection of MB coding modes and quantizer scales. In the Test Model (MPEG-2 Test model 5 [1993]), the rate control operates independently from the coding mode selection for each MB. The coding mode is decided based only upon the energy of predictive residues. Actually, the two processes, coding mode decision and rate control, are intimately related to each other and should be determined jointly in order to achieve optimal coding performance. A constrained optimization problem can be formulated based on the rate-distortion characteristics, or R(D) curves, for all the MBs that compose the picture being coded. Distortion for the entire picture is assumed to be decomposable and expressible as a function of individual MB distortions, with this being the objective function to minimize. The determination of the optimal solution is complicated by the MPEG differential encoding of motion vectors and dc coefficients, which introduce dependencies that carry over from MB to MB for a duration equal to the slice length. As an approximation, a near-optimum greedy algorithm can be developed. Once the upper bound in performance is calculated, it can be used to assess how well practical sub-optimum methods perform.
Analytical methods
Published in Charles Aubeny, Geomechanics of Marine Anchors, 2017
An upper bound analysis involves postulating a kinematically admissible collapse mechanism, from which a collapse load is computed by equating the internal rate of energy dissipation to the rate of work performed by external loads; that is, a virtual work analysis. The method has the advantage that the collapse mechanism can be systematically varied to seek an optimal mechanism producing a least upper bound. While the “least upper bound” for a presumed collapse mechanism is still not necessarily equal to the exact solution, the optimization process can nonetheless provide increased confidence in the result. As the calculations are often relatively straightforward, upper bound methods are readily adaptable to routine design calculations. Occasionally, the upper and lower bound collapse load estimates yield identical results, in which case the solution may be considered exact. Otherwise, we need to content ourselves with a bounded estimate of load capacity, which is nevertheless frequently useful.
The estimation of approximation error using inverse problem and a set of numerical solutions
Published in Inverse Problems in Science and Engineering, 2021
A. K. Alekseev, A. E. Bondarev
One may treat the norms of the exact error and estimate error as the radii of hyperspheres and in the space of discrete solutions. Then the relation means that the hypersphere, containing the projection of exact error onto grid, belongs to the hypersphere defined by the estimator. Thus, in order to provide a reliable estimation, the effectivity index should be greater than the unit. Also, the estimate should be not too pessimistic, so the value of the effectivity index should be not too great. For the finite element applications in the domain of elliptic equations (as usual, highly regular), the acceptable range of the effectivity index, according to [7], is . It should be noted that the upper bound is not strictly defined and may be problem dependent. The solutions, considered herein, contain discontinuities (shear lines, shock waves), so the acceptable range of the effectivity index may be greater and corresponding bounds should be established by an additional analysis (for example, from the acceptable errors of the valuable functionals, which are used in applications).