Concave Finite Element Shape Functions

凹有限元形函数

• 批准号: 0202232
• 负责人: Gautam Dasgupta
• 金额: --
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-06-01 至 2007-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202232&HistoricalAwards=false
• 关键词: Concave; Finite; Element; Shape; Functions

Geometrical concavity in finite elements does not permit the displacement-based shape functions that are restricted to algebraic polynomials. Functions with singularity and rational polynomials (numerator polynomials divided by denominator polynomials), which alleviate the limitation of Taylor polynomials, will be constructed to yield shape functions to guarantee a preassigned degree of continuity within a concave finite element. The application of exact integration (not conventional numerical quadrature schemes) based on the divergence theorem will be employed to integrate the energy density functions leading to element mass and stiffness matrices. In stress-based (hybrid) finite elements, concavity does not pose any additional problems. Theprocedures available for convex polygonal elements will be utilized with appropriate modifications to account for concavity. These novel finite elements are essential in high quality models that necessitate minimum number of tessellations. In addition to domain discretized procedures, three-dimensional boundary element models will utilize the proposed elements to cover boundaries where reentrant corners exist. As an outgrowth of this research, a limited number of curved boundaries, will be pursued to augment the element library. Conventional convex elements with limited forms of geometrical shapes, where isoparametric transformations are routinely employed, will be incorporated in the proposed numerical formulation as special cases.Fundamental mathematical results of projective and perspective geometrical considerations will guide the construction of shape functions including the singular non-classical ones in closed analytical forms. Computer algebra code will symbolically manipulate singular functions as distributions (generalized functions). The weak definition of the square root to generate absolute values, the delta sequences and their formal derivatives to construct Dirac's delta and Heaviside step functions will be seamlessly accommodated within the element shape function and integration routines. All integrals will be interpreted in the distributional sense in a non-classical (weak) formulation. Object-oriented complied codes will result from the computer algebraic constructs based on closed-form manipulations of analytical representations of singularities. This integrated computational environment, which consists of algebraic, numerical and graphics routines, will be employed to generate shape functions and system matrices on both convex and concave domains in order to solve large scale problems where high numerical accuracy cannot be compromised.Application fields will include geotechnical modeling for blasts, soil-structure interactions and bioengineering growth analysis of soft and hard tissues. Second order effects of randomness in constitutive descriptions and boundary geometry will be studied as benchmark examples. High precision applications in the field of high technology, e.g., optical determination of constitutive properties in the micro and lower level material structures, will be explored.

有限单元的几何凹性不允许基于位移的形状函数被限制为代数多项式。具有奇异多项式和有理多项式（分子多项式除以分母多项式）的函数将被构造成形状函数，以保证凹有限元中预先指定的连续性程度，从而减轻泰勒多项式的局限性。应用基于散度定理的精确积分（而不是传统的数值积分格式）对导致单元质量和刚度矩阵的能量密度函数进行积分。在基于应力的（混合）有限元中，凹性不会造成任何额外的问题。可用于凸多边形元素的程序将在适当修改后用于解释凹性。这些新颖的有限元素是必要的高质量的模型，需要最少数量的镶嵌。除了领域离散化程序，三维边界元素模型将利用所提出的元素来覆盖可重入角存在的边界。作为本研究的一个结果，将追求有限数量的弯曲边界来增加元素库。具有几何形状有限形式的传统凸元素，通常采用等参变换，将作为特殊情况纳入建议的数值公式中。投影和透视几何考虑的基本数学结果将指导形状函数的构造，包括封闭解析形式的奇异非经典函数。计算机代数代码将符号地将奇异函数作为分布（广义函数）来操作。生成绝对值的平方根的弱定义，构造狄拉克的delta和Heaviside阶函数的delta序列及其形式导数，将无缝地容纳在元素形状函数和积分例程中。所有的积分将在非经典（弱）公式的分布意义上解释。面向对象的编译代码将产生于基于对奇异点解析表示的封闭形式操作的计算机代数构造。这个集成的计算环境由代数、数值和图形例程组成，将用于在凸域和凹域上生成形状函数和系统矩阵，以解决大规模问题，在这些问题中，高数值精度不能受到损害。应用领域将包括爆炸的岩土建模、土壤-结构相互作用和软硬组织的生物工程生长分析。随机性在本构描述和边界几何中的二阶效应将作为基准例子进行研究。在高科技领域的高精度应用，例如，在微观和较低水平的材料结构的本构性质的光学测定，将被探索。