{'fraction_non_alphanumeric': 0.0383419689119171, 'fraction_numerical': 0.01606217616580311, 'mean_word_length': 5.4366666666666665, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A new five‐node locking‐free quadrilateral element based on smoothed FEM for near‐incompressible linear elasticity

The volumetric locking issue is critical in finite element analysis of nearly incompressible problems. In this paper, a new five‐node quadrilateral element (Q5) is proposed by enriching the four‐node quadrilateral element (Q4) with a centroid node to solve the volumetric locking problem in FEM. The cell‐based smoothed FEM is employed with Q5 element (referred as Q5‐CS‐SC4) to soften the stiffness in order to obtain a better solution. To eliminate pressure oscillation in near‐incompressible problems, an edge‐based area‐weighted smoothing scheme incorporated with the cell‐wise divergence‐free Q5 element is carried out (referred as Q5‐EAW), and an adjustable area‐weighted strain smoothing scheme using a parameter p is proposed to improve the performance of the Q5 element in dealing with incompressible media (referred as Q5‐pEAW). The formulation of Q5‐pEAW is a simple combination of Q5‐CS‐SC4 and Q5‐EAW by an adjustable area weight. It can search the exact strain energy of the problem in near‐incompressible cases. We also introduce another node‐based strain smoothing technique (Q5‐NAW) into the domain‐based selective scheme to obtain a Q5‐pEAW/NAW model to solve the pressure oscillation, which gives a much smoother pressure solution than Q5‐EAW. Finally, the Q5 element is extended into hexahedral element to develop a nine‐node hexahedral (H9) element shape function by enriching the eight‐node hexahedral element (Q8) with a centroid node. An H9‐Gi/NAW model similar to Q5‐pEAW/NAW is proposed by using H9 element to solve the 3D volumetric locking. A series of benchmark problems are provided to demonstrate that the proposed Q5‐pEAW/NAW for 2D plane strain problems, and H9‐Gi/NAW model for 3D cases are locking‐free for nearly incompressible problems. Copyright © 2014 John Wiley & Sons, Ltd.