报告主题：A Scaled Boundary Framework for Generating Bubble-Enriched Basis Functions and Its Application to Designing Stable u-p Elements

报告人：Chongmin Song教授（澳大利亚新南威尔士大学）

报告时间：2026年6月23日（周二）10：00

报告地点：91麻豆 （江宁校区）致高A105

主办单位：91麻豆 动力学与控制研究所

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报告简介：

A scaled boundary framework for generating basis functions over star-convex polygons is proposed. A sequence of Poisson’s equations with monomial inhomogeneous terms, whose solutions span a complete polynomial basis of a prescribed order, is considered. Using the scaled boundary finite element method, these equations are weakened along the boundary of the polygons using line elements of arbitrary orders, leading to a system of ordinary differential equations (ODEs). An efficient procedure is developed to obtain the basis functions by solving these ODEs. The resulting basis functions are naturally separated into two parts: a homogeneous solution associated only with the boundary nodes, and a particular solution consisting of bubble functions that vanish on the boundary. This semi-analytical scheme provides a systematic approach for constructing compatible bases for coupled field variables and for enriching intra-element approximations founded on the well-established solution procedure of ODEs.

This approach is applied to develop stable u-p elements that eliminate volumetric locking in nearly incompressible and truly incompressible problems. The displacement bases are derived by incorporating the chosen pressure bases in the inhomogeneous terms of the sequence of Poisson’s equations. No stabilization parameters are needed in the formulation. The stability of the proposed family of discontinuous-pressure and continuous-pressure polygon elements are evaluated by the inf–sup constants of single- and multiple-element patches. All proposed elements with exception of the 3-node element with discontinuous pressure are stable for practical problems. Numerical examples are presented to demonstrate their effectiveness and robustness.