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A unified approach to strict upper and lower bounds of quantities in linear elasticity based on constitutive relation error estimation. (English) Zbl 1423.74112

Summary: This paper presents a unified approach to acquiring strict upper and lower bounds of various quantities in linear elasticity. The key ingredient lies in a unified representation of linear quantities including displacement integrals, stress integrals and even pointwise quantities. With the unified representation, dual error analysis can be performed easily which results in a unified approximation and thereby a unified error representation via the primal-dual equivalence theorem. Then, the constitutive relation error (CRE) estimation featured with the ability to provide strict upper bound of global energy norm error is utilized and strict upper and lower bounds of the quantities are obtainable thereafter. Moreover, two extant bounding approaches to goal-oriented error estimation are analyzed and optimized, and as a result, optimal approximation and bounds are obtained. Numerical examples are studied to validate the proposed unified approach.

MSC:

74B05 Classical linear elasticity
74G45 Bounds for solutions of equilibrium problems in solid mechanics
74S05 Finite element methods applied to problems in solid mechanics
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