Extending CAS elements to remove shear and membrane locking from quadratic NURBS‐based discretizations of linear plane Timoshenko rods
Extending CAS elements to remove shear and membrane locking from quadratic NURBS‐based discretizations of linear plane Timoshenko rods
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DOI：
10.1002/nme.7257
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发表时间：
2023-05
影响因子：
2.9
通讯作者：
Mahmoud Golestanian;Hugo Casquero
中科院分区：
文献类型：
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作者：
Mahmoud Golestanian;Hugo Casquero
最近引入了连续假定应变 (CAS) 单元（Casquero 和 Golestanian。ComputMethods Appl Mech Eng. 2022；399:115354.），以消除二次 C1$$ {C}^1 $$ 线性平面弯曲基尔霍夫杆的基于连续 NURBS 的离散化中存在的膜锁定。在这项工作中，我们推广了 CAS 单元，以消除线性平面弯曲 Timoshenko 杆的基于二次 NURBS 的离散化中的剪切和膜锁定。 CAS 元素是一种假设的应变处理，使用线性拉格朗日多项式对结处的剪切应变和膜应变进行插值。因此，剪切应变和膜应变的单元间连续性得以保持。这项工作中考虑的数值实验表明，CAS 元件消除了由剪切和膜锁定引起的剪切力和膜力的寄生振荡。此外，当使用完全或简化积分的 CAS 单元时，位移、旋转和应力合力的收敛与长细比无关，最高可达 104$$ 1{0}^4 $$，而使用 NURBS 单元时，收敛高度依赖于长细比。我们将 CAS 单元的锁定处理应用于二次 C0$$ {C}^0 $$ 连续 NURBS，所得单元类型称为不连续假定应变 (DAS) 单元。 CAS 和 DAS 元素之间的比较表明，一旦锁定被正确删除，跨元素边界的 C1$$ {C}^1 $$ 连续性比跨元素边界的 C0$$ {C}^0 $$ 连续性具有更高的精度。最后，CAS 元素产生了一个简单的数值方案，与易于锁定的基于 NURBS 的 Galerkin 方法离散化相比，该方案不会增加任何显着的计算负担。
Continuous‐assumed‐strain (CAS) elements were recently introduced (Casquero and Golestanian. Comput Methods Appl Mech Eng. 2022; 399:115354.) to remove the membrane locking present in quadratic C1$$ {C}^1 $$ ‐continuous NURBS‐based discretizations of linear plane curved Kirchhoff rods. In this work, we generalize CAS elements to remove shear and membrane locking from quadratic NURBS‐based discretizations of linear plane curved Timoshenko rods. CAS elements are an assumed strain treatment that interpolates the shear and membrane strains at the knots using linear Lagrange polynomials. Consequently, the inter‐element continuity of the shear and membrane strains is maintained. The numerical experiments considered in this work show that CAS elements excise the spurious oscillations in shear and membrane forces caused by shear and membrane locking. Furthermore, when using CAS elements with either full or reduced integration, the convergence of displacements, rotations, and stress resultants is independent of the slenderness ratio up to 104$$ 1{0}^4 $$ while the convergence is highly dependent on the slenderness ratio when using NURBS elements. We apply the locking treatment of CAS elements to quadratic C0$$ {C}^0 $$ ‐continuous NURBS and the resulting element type is named discontinuous‐assumed‐strain (DAS) elements. Comparisons among CAS and DAS elements show that once locking is properly removed, C1$$ {C}^1 $$ continuity across element boundaries results in higher accuracy than C0$$ {C}^0 $$ continuity across element boundaries. Lastly, CAS elements result in a simple numerical scheme that does not add any significant computational burden in comparison with the locking‐prone NURBS‐based discretization of the Galerkin method.