An Extremal Problem Arising in the Dynamics of Two‐Phase Materials That Directly Reveals Information about the Internal Geometry
An Extremal Problem Arising in the Dynamics of Two‐Phase Materials That Directly Reveals Information about the Internal Geometry
复制标题
直接揭示内部几何信息的两相材料动力学中出现的极值问题
DOI：
10.1002/cpa.22082
复制
发表时间：
2022
影响因子：
3
通讯作者：
Putinar, Mihai
中科院分区：
文献类型：
--
作者：
Mattei, Ornella;Milton, Graeme W.;Putinar, Mihai
在两相材料中，每个相在时间上都有一个非局部响应，人们已经发现，对于一些驱动场，响应在特定的时间以某种方式解开，并允许人们直接推断关于材料几何的有用信息，例如相的体积分数。受此启发，为了得到设计合适的驱动场的算法，我们找到了马尔可夫函数在给定的可能复点的集合上所取的值之间的近似、度量独立的线性关系，而不属于该度量被支持的区间[-1，1]。将问题简化为给定函数在区间[-1，1]上的多项式逼近问题，并用切比雪夫逼近简化了分析。这使得人们可以根据点数和这些点到区间[-1，1]的最小距离来获得近似误差的显式估计。假设这个最小距离被一个大于1/2的数字限制在下面，随着点数的增加，误差以指数形式收敛到零。还得到了包含一组测量矩的近似线性关系。在激励问题的背景下，该分析还得出了任何驾驶场在任何特定时间的响应的界限，并允许人们使用适当设计的仅在固定时间间隔内有效开启的驾驶场来估计在给定频率下的响应。该近似直接推广到具有半正定算子值测度的马尔可夫型函数，并且当依赖于时间的边界势被适当地剪裁时，这对于从特定时刻的边界通量测量确定物体中夹杂的形状具有应用。©2022威利期刊公司。
In two‐phase materials, each phase having a non‐local response in time, it has been found that for some driving fields the response somehow untangles at specific times, and allows one to directly infer useful information about the geometry of the material, such as the volume fractions of the phases. Motivated by this, and to obtain an algorithm for designing appropriate driving fields, we find approximate, measure independent, linear relations between the values that Markov functions take at a given set of possibly complex points, not belonging to the interval [‐1,1] where the measure is supported. The problem is reduced to simply one of polynomial approximation of a given function on the interval [‐1,1] and, to simplify the analysis, Chebyshev approximation is used. This allows one to obtain explicit estimates of the error of the approximation, in terms of the number of points and the minimum distance of the points to the interval [‐1,1]. Assuming this minimum distance is bounded below by a number greater than 1/2, the error converges exponentially to zero as the number of points is increased. Approximate linear relations are also obtained that incorporate a set of moments of the measure. In the context of the motivating problem, the analysis also yields bounds on the response at any particular time for any driving field, and allows one to estimate the response at a given frequency using an appropriately designed driving field that effectively is turned on only for a fixed interval of time. The approximation extends directly to Markov‐type functions with a positive semidefinite operator valued measure, and this has applications to determining the shape of an inclusion in a body from boundary flux measurements at a specific time, when the time‐dependent boundary potentials are suitably tailored. © 2022 Wiley Periodicals, Inc.
登录
查看更多内容
弱马尔可夫系统的切比雪夫近似表征
影响因子：
3.7
作者：
C. Micchelli
通讯作者：
C. Micchelli
关于两相介质的有效粘弹性模量 II。三维复剪切模量的严格界限。
DOI：
--
发表时间：
1997
期刊：
Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences
影响因子：
--
作者：
G. Milton;J. Berryman
通讯作者：
J. Berryman
DOI：
--
发表时间：
2010
期刊：
影响因子：
--
作者：
S. Kajiura;A. Cornett;K. Yopak
通讯作者：
K. Yopak
复合介质的逆向传输问题
DOI：
--
发表时间：
1990
期刊：
影响因子：
--
作者：
R. McPhedran;G. Milton
通讯作者：
G. Milton
对科学中普遍存在的线性连续介质方程的统一观点。
DOI：
--
发表时间：
2020
期刊：
影响因子：
--
作者：
G. Milton
通讯作者：
G. Milton