Research Directions in Computational Mechanics (1991) / Chapter Skim
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1 Adaptive Methods and Error Estimation
1 Adaptive Methods and Error Estimation
Pages 17-25
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From page 17... ...
, density of gridpoints, and order of the representation. It is clear that in this necessary first step of the computational process, the discretization, an error is always made, because a discrete model cannot possibly capture all of the information embodied in continuum models of gaseous, fluid, or solid materials.
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From page 18... ...
Typically, finite element methods, finite difference methods, or boundary element methods are used in the discretization process, and the error is measured by error indicators that express the error in a single element or gridcell in an appropriate norm. Numerical experiments are generally performed on benchmark problems for which the exact error can be calculated, and the effectiveness of the error estimate can be assessed by computing effectively indices defined by i= lest i m a tech error true error Today, most of the a posterior)
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From page 19... ...
Additional work is needed on error estimation procedures for boundary element methods, finite difference methods, spectral and spectral element schemes, and other related techniques. For the short term, research will continue to focus on models characterized by linear elliptic equations, but research on the important areas of time-dependent and nonlinear problems will be vital to future developments in adaptive methods throughout the next decade.
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From page 20... ...
Theoretically, in many cases adaptive procedures can lead to exponential convergence rates compared to algebraic convergence of classical low-order nonadaptive approaches. The potential impact on computational mechanics of exponentially convergent computational schemes may be one of the most important factors affecting research in this area for the next decade.
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From page 21... ...
Ad a ptive Method s in' 21 FIGURE 1.1 a An in-adapted finite element mesh about a shuttle-like body. Note that the adaptive algorithm has automatically refined the mesh to capture shocks in the flow field.
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22 RESEARCH DIRECTIONS 1 FIGURE l.lb Computed pressure contours on the optimally refined mesh.
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FIGURE 1.2 Numerical performance of various classes of numerical methods: logarithm of error versus logarithm of the number N of unknowns.
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To date, certain p-version adaptive finite element methods, combined h-p adaptive finite element methods, and spectral, pseudospectral, and spectral element methods have attained exponential convergence rates. Further research on adaptive strategies that produce optimal distributions of discretization parameters and exponential rates of convergence is urgently needed and should be a principal topic of research during the next decade.
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From page 25... ...
error estimation techniques in fluid mechanics for both compressible and incompressible f low; explore and develop a posterior) estimates and adaptive methods for space-time approximations; determine optimal techniques for h-, p-, and r-adaptive schemes; develop a modeling reliability theory, including possible expert system development and bracketing theory; devise postprocessing techniques for enhancement of solution accuracy; and develop adaptive modeling in which criteria for changes of models are incorporated in the adaptive process.
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