Locking-free isogeometric discretizations of linear plane Timoshenko rods: LAS elements
Locking-free isogeometric discretizations of linear plane Timoshenko rods: LAS elements
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DOI：
10.1016/j.cma.2024.116918
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发表时间：
2024-05
影响因子：
7.2
通讯作者：
Md Sadman Faruque;Hugo Casquero
中科院分区：
文献类型：
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作者：
Md Sadman Faruque;Hugo Casquero
我们利用线性平面弯曲的Timoshenko棒作为模型问题来研究如何克服基于nurbs的离散化中的剪切和膜锁定问题。在这项工作中，我们提出了集总假设应变（LAS）单元，这是一种基于投影的假设应变处理，可以在非常广泛的长细比范围内消除剪切和膜锁定。对于具有p次和cp−1次跨单元边界连续基函数的NURBS补片，假设应变的空间定义在具有相同单元的b样条补片上，但使用p−1次和cp−2次跨单元边界连续基函数。假设的菌株是通过在贴片水平上对相容菌株进行l2投影得到的，其中一致质量矩阵被集中质量矩阵取代。我们的数值研究表明，为了使未知数的l2范数的相对误差和应力结果都低于1%，LAS单元需要与B单元相同或稍细的网格分辨率。然而，对于给定的网格，LAS元素的计算效率明显高于B ^元素。使用c1 -连续二次LAS单元，每个单元有2个高斯-勒让德正交点，对大多数长细比显示应力结果的超收敛，使其成为获得准确结果的特别经济有效的选择。与每个单元具有2个高斯-勒让德正交点的线性拉格朗日B元相比，每个单元具有2个高斯-勒让德正交点的c1 -连续二次LAS单元所需的单元数量要少得多，从而获得未知数l2范数的相对误差和应力结果小于1%。
We make use of linear plane curved Timoshenko rods as a model problem to study how to overcome shear and membrane locking in NURBS-based discretizations. In this work, we propose lumped-assumed-strain (LAS) elements, a projection-based assumed-strain treatment that removes shear and membrane locking for a very broad range of slenderness ratios. For a NURBS patch with basis functions of degree p and C p− 1 continuity across element boundaries, the space of the assumed strains is defined on a B-spline patch with the same elements, but using basis functions of degree p− 1 and C p− 2 continuity across element boundaries. The assumed strains are obtained by performing a L 2 projection of the compatible strains at the patch level in which the consistent mass matrix is substituted with the lumped mass matrix. Our numerical investigations suggest that LAS elements need either the same or slightly finer mesh resolutions than B ̄ elements in order for the relative errors in L 2 norm of the unknowns and the stress resultants to be all below 1%. However, LAS elements are significantly more computationally efficient than B ̄ elements for a given mesh. The use of C 1-continuous quadratic LAS elements with 2 Gauss–Legendre quadrature points per element exhibit superconvergence of the stress resultants for most slenderness ratios, making it a particularly cost-effective choice of obtaining accurate results. When compared with linear Lagrange B ̄ elements with 2 Gauss–Legendre quadrature points per element, C 1-continuous quadratic LAS elements with 2 Gauss–Legendre quadrature points per element require far fewer elements in order to obtain relative errors in L 2 norm of the unknowns and the stress resultants smaller than 1%.