再生光滑梯度积分高效高阶无网格法

Higher order meshfree methods fully reflect the higher order smoothing and conforming approximation property and superior accuracy associated with meshfree methods, thus they are particularly desirable in the field of structural analysis. Nonetheless, the low computational efficiency issue becomes much more prominent for higher order meshfree methods, which strongly prevents the developments and applications of this class of methods. In this project, a new reproducing kernel gradient smoothing technique is firstly proposed to directly construct smoothed gradients for arbitrary order meshfree shape functions. Subsequently an algorithm is presented to optimize the integration sampling points for the smoothed gradient construction, which is capable of significantly accelerating meshfree computation with higher order shape functions. Thus efficient higher order meshfree methods based upon the reproducing kernel gradient smoothing integration technique are rationally proposed. The proposed methodology builds the higher order integration constraints into the reproducing kernel gradient smoothing process and the resulting smoothed gradients of meshfree shape functions automatically meet the integration constraints and completeness conditions. Meanwhile, the property of boundary sampling points shared by neighboring integration subdomains are explored to optimize the numbers and locations of integration sampling points. This integration optimization procedure ensures the computational accuracy and remarkably improves the computational efficiency of higher order meshfree methods. It is also noted that the proposed reproducing kernel smoothed gradients of meshfree shape functions are not restricted to discrete integration points and continuously meet the integration constraints throughout the problem domain, which have a similar structure as the meshfree shape functions and thus are very easy for numerical implementation. Arbitrary order multidimensional formulations of the proposed methodology is straightforward as well. The present higher order meshfree methods with reproducing kernel gradient smoothing integration can provide a very efficient numerical tool for superiorly accurate structural analysis.

高阶无网格法彰显了无网格形函数整体协调光滑和计算精度高的优点，在结构分析领域有重要应用价值，但计算效率低的问题尤为突出，已成为其发展和应用的瓶颈。本项目拟研究任意阶无网格形函数的再生光滑梯度直接构造方法，发展再生光滑梯度数值积分采样点选取的优化方案，进而建立再生光滑梯度积分高效高阶无网格法。该方法将高阶积分约束条件内嵌于形函数再生光滑梯度构造过程，使得再生光滑梯度自然满足高阶积分约束条件与完备性条件；充分利用再生光滑梯度数值积分采样点在背景积分域间的共享特性，在保证无网格法数值积分精度的前提下优化形函数再生光滑梯度构造过程中积分采样点的数量和位置，能够显著提高高阶无网格法的计算效率；形函数再生光滑梯度不受限于离散积分点，具有与无网格形函数相似的构造形式并在全域内满足积分约束条件，数值实现简洁，适用于任意高阶和多维无网格法。再生光滑梯度积分高阶无网格法可为结构分析提供一种高效的高精度方法。

本项目提出了任意阶伽辽金无网格法的数值积分精度度量方法，并通过优化积分采样点布置，发展了无网格形函数再生光滑梯度的直接构造理论和方法，进而建立了再生光滑梯度积分高效高阶伽辽金无网格法。该方法内嵌了局部积分约束条件，因而自然满足全域积分约束条件，保证了方法的最优收敛特性。同时，再生光滑梯度具有显式表达式，计算高效，数值实现简捷。此外，本项目研究了无网格配点法的精度度量理论，发展了基于递推光滑梯度的超收敛无网格配点法，为解决配点法采用奇数阶次基函数时精度掉阶问题提供了一种有效途径。再者，本项目还提出了一种弹性体频率的超收敛等几何分析方法，显著提升了弹性体频率计算精度。项目执行期内发表期刊论文25篇，包括计算力学领域主流期刊Computer Methods in Applied Mechanics and Engineering 5篇，Computational Mechanics 4篇、力学学报2篇。项目负责人获杜庆华工程计算方法奖和Emerald高度推荐论文奖，并入选爱思维尔中国高被引学者。

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