Concave Finite Element Shape Functions

几何拟合 有限元不允许基于位移的形状函数被限制为代数多项式。将构造具有奇异性和有理多项式（分子多项式除以分母多项式）的函数，以产生形函数，从而保证凹有限元内的预定连续度，从而减轻泰勒多项式的限制。将采用基于发散定理的精确积分（而不是传统的数值求积方案）对能量密度函数进行积分，从而得到单元质量和刚度矩阵。在基于应力的（混合）有限元中，不确定性不会带来任何额外的问题。凸多边形单元的程序将在适当的修改后加以利用，以考虑凸多边形的影响。这些新的有限元是必不可少的高品质的模型，需要最少数量的镶嵌。除了区域离散化程序，三维边界元模型将利用建议的元素，以涵盖边界处的凹角存在。 作为这项研究的一个结果，有限数量的弯曲边界，将追求扩大单元库。传统的凸元素与有限形式的几何形状，其中等参变换是例行采用，将被纳入拟议的数值formulation作为特殊cases.Fundamental数学结果的投影和透视几何考虑将指导建设的形状函数，包括奇异的非经典的封闭的解析形式。计算机代数代码将把奇异函数作为分布（广义函数）进行符号化处理。用于生成绝对值的平方根的弱定义、用于构造狄拉克δ和海维赛德阶跃函数的δ序列及其形式导数将无缝地容纳在单元形状函数和积分例程中。所有的积分将在非经典（弱）公式中的分布意义上解释。面向对象的编译代码将导致从计算机代数结构的基础上封闭形式的操作的分析表示的奇点。 这个综合计算环境包括代数、数值和图形程序，将用于在凸域和凹域上生成形函数和系统矩阵，以解决高数值精度的大规模问题，应用领域包括爆炸的岩土模拟、土壤-结构相互作用和软硬组织的生物工程生长分析。 本构描述和边界几何形状的随机性的二阶效应将作为基准例子进行研究。高科技领域的高精度应用，例如，光学测定的本构性质的微观和较低层次的材料结构，将探讨。

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.