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Parametric finite-volume method for Saint Venant's torsion of arbitrarily shaped cross sections

机译：圣门的扭曲任意形状横截面扭转的参数有限体积法

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We extend our recent finite-volume based approach to the solution of Saint Venant's torsion problems of bars and beams comprised of rectangular sections to enable analysis of arbitrary cross sections characterized by curved boundaries. This is accomplished by incorporating parametric mapping based on transfinite grid generation to enable discretization of the bar cross section by quadrilateral rather than rectangular subvolumes employed in the original version. The construction of the local stiffness matrix that relates the surface averaged subvolume warping functions to the corresponding tractions is carried out in the reference plane such that the subvolume equilibrium in the physical plane is satisfied in a surface-averaged sense. This produces explicit expressions for the stiffness matrix elements that may be readily coded. Orthotropic subvolumes are intrinsic in the method's construction so that bars with heterogeneous and composite microstructures may be analyzed. The convergence and accuracy of the parametric finite-volume method are assessed and verified upon comparison with exact elasticity solutions for cross sections with convex and concave boundaries. Examples involving structural applications of prismatic bars with curved boundaries illustrate the utility of the developed methodology. These include cross sections that resemble biological constructs with homogeneous and graded regions aimed at enhancing torsional rigidities, as well as homogeneous and graded elliptical cross sections with orthotropic shear moduli aimed at reducing and eliminating warping. We demonstrate for the first time that by laminating an elliptical cross section with alternating stiff and soft isotropic layers in a manner that mimics the required orthotropic moduli at the homogenized level, warping can be practically eliminated with sufficient microstructural refinement.
机译：我们将最近的基于有限卷的方法扩展到圣文尼的磁条的扭转问题的解决方案，包括矩形部分，以便能够分析特征的弯曲边界的任意横截面。这是通过基于Transfinite网格生成的参数映射来实现的，以使得Bar横截面通过在原始版本中采用的四边形而不是矩形子域来实现条形横截面的离散化。局部平均子荚膜翘曲与相应牵引的局部刚度矩阵的构造在参考平面中进行，使得在表面平均意义上满足物理平面中的子培养平衡。这为可以容易地编码的刚度矩阵元件产生显式表达式。正交亚伏在该方法的结构中是内在的，因此可以分析具有异质和复合微结构的条。在与具有凸形和凹形边界的横截面的精确弹性溶液比较，评估和验证参数有限体积法的收敛性和准确性。涉及具有弯曲边界的棱柱杆的结构应用的示例说明了开发方法的效用。这些包括与均匀的和渐变区域类似于增强扭转刚性的生物构建体的横截面，以及具有正向剪切模量的均匀和分级椭圆形横截面，其旨在减少和消除翘曲。我们首次证明了通过以交替的刚性和柔软的各向同性层层叠椭圆形横截面，以使所需的正向调制在均质水平下的所需的正向调制性的方式，可以通过足够的微观结构细化实际消除翘曲。

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