Quantifying the Impact of Material-Model Error on Macroscale Quantities-of-Interest Using Multiscale a Posteriori Error-
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Quantifying the Impact of Material-Model Error on Macroscale Quantities-of-Interest Using Multiscale a Posteriori Error-Estimation Techniques Judith A. Brown1 and Joseph E. Bishop1 1
Engineering Sciences Center, Sandia National Laboratories, Albuquerque, NM 87185, U.S.A.
ABSTRACT An a posteriori error-estimation framework is introduced to quantify and reduce modeling errors resulting from approximating complex mesoscale material behavior with a simpler macroscale model. Such errors may be prevalent when modeling welds and additively manufactured structures, where spatial variations and material textures may be present in the microstructure. We consider a case where a fiber texture develops in the longitudinal scanning direction of a weld. Transversely isotropic elastic properties are obtained through homogenization of a microstructural model with this texture and are considered the reference weld properties within the error-estimation framework. Conversely, isotropic elastic properties are considered approximate weld properties since they contain no representation of texture. Errors introduced by using isotropic material properties to represent a weld are assessed through a quantified error bound in the elastic regime. An adaptive error reduction scheme is used to determine the optimal spatial variation of the isotropic weld properties to reduce the error bound. INTRODUCTION Two fundamental sources of error in macroscale solid-mechanics modeling are (1) the assumption of a separation-of-scales in homogenization theory and (2) the use of macroscopic material models that attempt to represent behavior governed by microscale processes. These errors may be large when modeling welds, where the local microstructure has unique features governed by solidification of the melt pool. The resulting morphology depends on the ratio of the melt pool temperature gradients to the local solidification rate [1]. This is governed by processing parameters such as scan speed [2] and heat input [3]. Additionally, a fiber-like texture with the crystal directions oriented along the solidification direction of the grains has been observed in both welds [1,3] and additively manufactured materials [2,4-5]. On engineering length scales, welds are often approximated by simplified material models that assume homogeneous, isotropic properties throughout the structure and thus neglect the effects of texture. In order to quantify these approximation errors, we adopt the modeling error-estimation framework proposed by Oden et al. [6] to study the response of a stainless steel autogenous welded structure. We use this approach to evaluate errors introduced by using a homogeneous, isotropic material model to represent welded portions of a structure compared with a higher-fidelity model that considers the effects of weld texture. THEORY We use concepts from the field of a posteriori error estimation to bound the error arising from the use of an approximate material model [7-11]. Here, we focus on quantifying the error in the global energy norm of the di
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