Abstract This paper introduces a novel method to model non-deterministic quantities based on experimental measurement data. The focus of this work is on quantities that vary over a continuous domain, e.g., material properties, time-dependent strain rate effects, or stress–strain curves. These quantities are modelled by means of the recently introduced concept of interval fields. An interval field defines intervals that are defined throughout the continuous domain and have dependence in this domain by expanding them over a set of basis functions, describing the spatial nature of the non-determinism of the modelled quantities. One of the more intuitive concepts of defining basis functions in an interval field is through inverse distance weighting interpolation (IDW), which starts from known intervals at specific control points within the domain. For each of these control points, a corresponding basis function is defined, the relative weight of which is decreasing inversely with the distance. Through this definition, all intervals have non-vanishing basis functions throughout the model domain. This makes the application of standard IDW extremely challenging when the interval uncertainty varies inhomogeneously over the domain, i.e., when local effects are present in the model. Therefore, in this paper standard IDW is adapted by changing the distance measure. More specifically, the weight of intervals is increased locally, while diminishing the weight in other regions. For this purpose, a function is introduced that maps the domain to a higher dimension feature space, in which the distances that determine the weight are measured. This mapping function is based on either the size of the intervals at the control points or experimental data, which both yield additional control resulting in increased agreement with experimental data. This paper demonstrates that this method outperforms standard IDW in controllability, while limiting the number of control points. This is illustrated in three case studies: a first case concerning modelling local non-determinism; a second case where a mix of global and local effects is modelled; and the third case, where the interval field is based on experimental stress strain curves. In all these cases, multiple configurations demonstrate the effects of the parameters, and how the new technique is applied. The proposed technique outperforms standard IDW in all three case studies, with an increased coefficient of determination, R 2 , between 22% and 56%, in comparison to standard IDW.

中文翻译：

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基于比例反距离加权插值的非齐次区间场

摘要 本文介绍了一种基于实验测量数据对非确定性量进行建模的新方法。这项工作的重点是在连续域上变化的量，例如材料特性、瞬态应变率效应或应力-应变曲线。这些量是通过最近引入的区间场概念建模的。区间字段定义了在整个连续域中定义的区间，并且通过在一组基函数上扩展它们在该域中具有相关性，描述了建模量的非确定性的空间性质。在区间域中定义基函数的一个更直观的概念是通过反距离加权插值 (IDW)，它从域内特定控制点的已知区间开始。对于这些控制点中的每一个，定义了相应的基函数，其相对权重与距离成反比减小。通过这个定义，所有区间在整个模型域中都具有非零基函数。当区间不确定性在域上不均匀变化时，即当模型中存在局部效应时，这使得标准 IDW 的应用极具挑战性。因此，本文通过改变距离度量来适应标准 IDW。更具体地说，区间的权重在局部增加，而其他区域的权重减小。为此，引入了一个函数，将域映射到更高维度的特征空间，在其中测量确定权重的距离。此映射函数基于控制点的间隔大小或实验数据，这两者都会产生额外的控制，从而增加与实验数据的一致性。本文证明该方法在可控性方面优于标准 IDW，同时限制了控制点的数量。这在三个案例研究中得到了说明：第一个案例涉及建模局部非确定性；第二种情况，对全球和局部效应的混合进行建模；第三种情况，其中区间场基于实验应力应变曲线。在所有这些情况下，多个配置展示了参数的影响，以及如何应用新技术。所提出的技术在所有三个案例研究中都优于标准 IDW，增加了决定系数 R 2 ，