Bound Solutions (bound + solution)

Distribution by Scientific Domains
Engineering100%

Kinds of Bound Solutions

  • upper bound solution
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    Selected Abstracts

    Certified solutions for hydraulic structures using the node-based smoothed point interpolation method (NS-PIM)

    INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 15 2010
    J. Cheng
    Abstract A meshfree node-based smoothed point interpolation method (NS-PIM), which has been recently developed for solid mechanics problems, is applied to obtain certified solutions with bounds for hydraulic structure designs. In this approach, shape functions for displacements are constructed using the point interpolation method (PIM), and the shape functions possess the Kronecker delta property and permit the straightforward enforcement of essential boundary conditions. The generalized smoothed Galerkin weak form is then applied to construct discretized system equations using the node-based smoothed strains. As a very novel and important property, the approach can obtain the upper bound solution in energy norm for hydraulic structures. A 2D gravity dam problem and a 3D arch dam problem are solved, respectively, using the NS-PIM and the simulation results of NS-PIM are found to be the upper bounds. Together with standard fully compatible FEM results as a lower bound, we have successfully determined the solution bounds to certify the accuracy of numerical solutions. This confirms that the NS-PIM is very useful for producing certified solutions for the analysis of huge hydraulic structures. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    Lower bound limit analysis with adaptive remeshing

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 14 2005
    Andrei V. Lyamin
    Abstract The objective of this work is to present an adaptive remeshing procedure for lower bound limit analysis with application to soil mechanics. Unlike conventional finite element meshes, a lower bound grid incorporates statically admissible stress discontinuities between adjacent elements. These discontinuities permit large stress jumps over an infinitesimal distance and reduce the number of elements needed to predict the collapse load accurately. In general, the role of the discontinuities is crucial as their arrangement and distribution has a dramatic influence on the accuracy of the lower bound solution (Limit Analysis and Soil Plasticity, 1975). To ensure that the discontinuities are positioned in an optimal manner requires an error estimator and mesh adaptation strategy which accounts for the presence of stress singularities in the computed stress field. Recently, Borges et al. (Int. J. Solids Struct. 2001; 38:1707,1720) presented an anisotropic mesh adaptation strategy for a mixed limit analysis formulation which used a directional error estimator. In the present work, this strategy has been tailored to suit a discontinuous lower bound formulation which employs the stresses and body forces as primary unknowns. The adapted mesh has a maximum density of discontinuities in the direction of the maximum rate of change in the stress field. For problems involving strong stress singularities in the boundary conditions (e.g. a strip footing), the automatic generation of discontinuity fans, centred on the singular points, has been implemented. The efficiency of the proposed technique is demonstrated by analysis of two classical soil mechanics problems; namely the bearing capacity of a rigid strip footing and the collapse of a vertical cut. Copyright © 2005 John Wiley & Sons, Ltd. [source]

    2D and 3D upper bound solutions for tunnel excavation using ,elastic' flow fields

    INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 12 2007
    Assaf Klar
    Abstract This paper presents 2D and 3D upper bound solutions for the problem of tunnel excavation in soft ground. The solution invokes the use of incompressible flow fields derived from the theory of elasticity and the concept of sinks and sources. Comparison is made with previously published results. For some geometries the current calculation results in lower (better) upper bound values; however, the results were generally close to previously published values. Copyright © 2007 John Wiley & Sons, Ltd. [source]

    Upper and lower bounds for natural frequencies: A property of the smoothed finite element methods

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 2 2010
    Zhi-Qian Zhang
    Abstract Node-based smoothed finite element method (NS-FEM) using triangular type of elements has been found capable to produce upper bound solutions (to the exact solutions) for force driving static solid mechanics problems due to its monotonic ,soft' behavior. This paper aims to formulate an NS-FEM for lower bounds of the natural frequencies for free vibration problems. To make the NS-FEM temporally stable, an ,-FEM is devised by combining the compatible and smoothed strain fields in a partition of unity fashion controlled by ,,[0, 1], so that both the properties of stiff FEM and the monotonically soft NS-FEM models can be properly combined for a desired purpose. For temporally stabilizing NS-FEM, , is chosen small so that it acts like a ,regularization parameter' making the NS-FEM stable, but still with sufficient softness ensuring lower bounds for natural frequency solution. Our numerical studies demonstrate that (1) using a proper ,, the spurious non-zero energy modes can be removed and the NS-FEM becomes temporally stable; (2) the stabilized NS-FEM becomes a general approach for solids to obtain lower bounds to the exact natural frequencies over the whole spectrum; (3) ,-FEM can even be tuned for obtaining nearly exact natural frequencies. Copyright © 2010 John Wiley & Sons, Ltd. [source]

    A novel singular node-based smoothed finite element method (NS-FEM) for upper bound solutions of fracture problems

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 11 2010
    G. R. Liu
    Abstract It is well known that the lower bound to exact solutions in linear fracture problems can be easily obtained by the displacement compatible finite element method (FEM) together with the singular crack tip elements. It is, however, much more difficult to obtain the upper bound solutions for these problems. This paper aims to formulate a novel singular node-based smoothed finite element method (NS-FEM) to obtain the upper bound solutions for fracture problems. In the present singular NS-FEM, the calculation of the system stiffness matrix is performed using the strain smoothing technique over the smoothing domains (SDs) associated with nodes, which leads to the line integrations using only the shape function values along the boundaries of the SDs. A five-node singular crack tip element is used within the framework of NS-FEM to construct singular shape functions via direct point interpolation with proper order of fractional basis. The mix-mode stress intensity factors are evaluated using the domain forms of the interaction integrals. The upper bound solutions of the present singular NS-FEM are demonstrated via benchmark examples for a wide range of material combinations and boundary conditions. Copyright © 2010 John Wiley & Sons, Ltd. [source]

    A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part II applications to solid mechanics problems

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 9 2010
    G. R. Liu
    Abstract In part I of this paper, we have established the G space theory and fundamentals for W2 formulation. Part II focuses on the applications of the G space theory to formulate W2 models for solid mechanics problems. We first define a bilinear form, prove some of the important properties, and prove that the W2 formulation will be spatially stable, and convergent to exact solutions. We then present examples of some of the possible W2 models including the SFEM, NS-FEM, ES-FEM, NS-PIM, ES-PIM, and CS-PIM. We show the major properties of these models: (1) they are variationally consistent in a conventional sense, if the solution is sought in a proper H space (compatible cases); (2) They pass the standard patch test when the solution is sought in a proper G space with discontinuous functions (incompatible cases); (3) the stiffness of the discretized model is reduced compared with the finite element method (FEM) model and possibly to the exact model, allowing us to obtain upper bound solutions with respect to both the FEM and the exact solutions and (4) the W2 models are less sensitive to the quality of the mesh, and triangular meshes can be used without any accuracy problems. These properties and theories have been confirmed numerically via examples solved using a number of W2 models including compatible and incompatible cases. We shall see that the G space theory and the W2 forms can formulate a variety of stable and convergent numerical methods with the FEM as one special case. Copyright © 2009 John Wiley & Sons, Ltd. [source]

    Lower bound limit analysis of cohesive-frictional materials using second-order cone programming

    INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 4 2006
    A. Makrodimopoulos
    Abstract The formulation of limit analysis by means of the finite element method leads to an optimization problem with a large number of variables and constraints. Here we present a method for obtaining strict lower bound solutions using second-order cone programming (SOCP), for which efficient primal-dual interior-point algorithms have recently been developed. Following a review of previous work, we provide a brief introduction to SOCP and describe how lower bound limit analysis can be formulated in this way. Some methods for exploiting the data structure of the problem are also described, including an efficient strategy for detecting and removing linearly dependent constraints at the assembly stage. The benefits of employing SOCP are then illustrated with numerical examples. Through the use of an effective algorithm/software, very large optimization problems with up to 700 000 variables are solved in minutes on a desktop machine. The numerical examples concern plane strain conditions and the Mohr,Coulomb criterion, however we show that SOCP can also be applied to any other problem of lower bound limit analysis involving a yield function with a conic quadratic form (notable examples being the Drucker,Prager criterion in 2D or 3D, and Nielsen's criterion for plates). Copyright © 2005 John Wiley & Sons, Ltd. [source]