A unified approach to strict upper and lower bounds of quantities in linear elasticity based on constitutive relation error estimation

• A unified representation of various quantities is presented in this paper.
• Even pointwise quantities are shown to fit well with the unified representation.
• Dual error estimation is conducted based on the unified representation.
• Two bounding approaches are revisited and optimized for the goal-oriented error estimation.
• Strict upper and lower bounds of displacement integrals, stress integrals and pointwise quantities are acquired by the present unified approach.

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.