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1. ANCF curvature continuity: application to soft and fluid materials

   https://doi.org/10.1007/s11071-020-05550-5  

   Shabana, Ahmed A.; Zhang, Dayu (April 2020, Nonlinear Dynamics)

   The continuity of the position-vector gradients at the nodal points of a finite element mesh does not always ensure the continuity of the gradients at the element interfaces. Discontinuity of the gradients at the interface not only adversely affects the quality of the simulation results, but can also lead to computer models that do not properly represent realistic physical system behaviors, particularly in the case of soft and fluid material applications. In this study, the absolute nodal coordinate formulation (ANCF) finite elements are used to define general curvature-continuity conditions that allow for eliminating or minimizing the discontinuity of the position gradients at the element interface. For the ANCF solid element, with four-node surfaces, it is shownthat continuity of the gradients tangent to an arbitrary point on a surface is ensured as the result of the continuity of the gradients at the nodal points. The general ANCF continuity conditions are applicable to both reference-configuration straight and curved geometries. These conditions are formulated without the need for using the computer-aided-design knot vector and knot multiplicity, which do not account properly for the concept of system degrees of freedom. The ANCF curvature-continuity conditions are written in terms of constant geometric coefficients obtained using the matrix of position-vector gradients that defines the reference-configuration geometry. The formulation of these conditions is demonstrated using the ANCF fully parameterized three-dimensional solid and tetrahedral elements, which employ a complete set of position gradients as nodal coordinates. Numerical results are presented in order to examine the effect of applying the curvature-continuity conditions on achieving a higher degree of smoothness at the element interfaces in the case of soft and fluid materials. 

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2. Constrained Large-Displacement Thermal Analysis

   https://doi.org/10.1115/1.4056182  

   Shabana, Ahmed A.; Elbakly, Mahmoud; Zhang, Dayu (February 2023, Journal of Computational and Nonlinear Dynamics)

   Abstract Two different cases are encountered in the thermal analysis of solids. In the first case, continua are not subject to boundary and motion constraints and all material points experience same displacement-gradient changes as the result of application of thermal loads. In this case, referred to as unconstrained thermal expansion, the thermal load produces uniform stress-free motion within the continuum. In the second case, point displacements due to boundary and motion constraints are restricted, and therefore, continuum points do not move freely when thermal loads are applied. This second case, referred to as constrained thermal expansion, leads to thermal stresses and its study requires proper identification of the independent coordinates which represent expansion degrees-of-freedom. To have objective evaluation and comparison between the two cases of constrained and unconstrained thermal expansion, the reference-configuration geometry is accurately described using the absolute nodal coordinate formulation (ANCF) finite elements. ANCF position-gradient vectors have unique geometric meanings as tangent to coordinate lines, allowing systematic description of the two different cases of unconstrained and constrained thermal expansions using multiplicative decomposition of the matrix of position-gradient vectors. Furthermore, generality of the approach for large-displacement thermal analysis requires using the Lagrange–D'Alembert principle for proper treatment of algebraic constraint equations. Numerical results are presented to compare two different expansion cases, demonstrate use of the new approach, and verify its results by comparing with conventional finite element (FE) approaches. 

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3. Well-balanced any order numerical solvers for linear and nonlinear poroelasticity based on combination of BDF and weak Galerkin

   https://doi.org/10.1093/imanum/drag020  

   Liu, Jiangguo; Tavener, Simon; Wang, Ruishu; Wang, Zhuoran (June 2026, IMA Journal of Numerical Analysis)

   Abstract This paper develops a family of any order finite element solvers for both linear and nonlinear poroelasticity problems. The backward differentiation formulas (BDFs) are used for temporal discretizations, whereas the weak Galerkin (WG) finite elements on quadrilaterals are utilized for spatial discretizations. For the latter, the discrete weak gradients of shape functions are established in the vector- or matrix- Arbogast–Correa spaces. Such combinations of BDF and WG discretizations produce numerical solutions that have well-balanced spatial and temporal errors in all six quantities, namely, displacement, dilation (divergence of displacement), stress, pressure, velocity and normal flux. For nonlinear poroelasticity problems in which permeability depends on solid dilation or the mean stress, Picard iterations are employed to solve the resulting nonlinear algebraic systems. Rigorous analysis together with numerical experiments demonstrate that these new solvers have optimal-order convergence and are free of locking. 

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4. Convergence of Lagrange finite elements for the Maxwell eigenvalue problem in two dimensions

   https://doi.org/10.1093/imanum/drab104  

   Boffi, Daniele; Guzmán, Johnny; Neilan, Michael (February 2022, IMA Journal of Numerical Analysis)

   Abstract We consider finite element approximations of the Maxwell eigenvalue problem in two dimensions. We prove, in certain settings, convergence of the discrete eigenvalues using Lagrange finite elements. In particular, we prove convergence in three scenarios: piecewise linear elements on Powell–Sabin triangulations, piecewise quadratic elements on Clough–Tocher triangulations and piecewise quartics (and higher) elements on general shape-regular triangulations. We provide numerical experiments that support the theoretical results. The computations also show that, on general triangulations, the eigenvalue approximations are very sensitive to nearly singular vertices, i.e., vertices that fall on exactly two ‘almost’ straight lines. 

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5. Phase 1: Simulating X65M Pipe Heat-Affected Zone Microstructures in Single Edge Notch Bending Toughness Samples to Assess Hydrogen Embrittlement

   https://doi.org/10.1115/PVP2025-154282  

   McCloskey, Jackson; Ni, Chen; Yu, Zhenzhen; Connolly, Matthew; Martin, May Ling; Acuna, Andres; Ebel, Greg; Bundy, Joe; Penso, Jorge; Menon, Ravi; et al (July 2025, American Society of Mechanical Engineers)

   Abstract Significant variations in the hydrogen embrittlement behavior were observed in the heat-affected zones (HAZ) of pipeline steels after fusion welding. Therefore, a thorough study is needed on the characteristics of subzone microstructures within HAZs of pipeline steels, especially after multi-pass welding, and their individual impacts on hydrogen embrittlement. This work employed a two-step methodology: (1) physical simulation of representative HAZ subzone microstructures into bulk specimens using a Gleeble® 3500 following thermal profiles obtained from finite element (FE) models, followed by (2) single edge notch bending (SE(B)) toughness tests under high pressure gaseous H2. This Phase I manuscript focuses on the physical microstructural simulation methodology. The SE(B) sample geometry requires a large cross-section dimension of 7 mm by 16 mm. The crack deviation from centerline of SE(B) specimens was observed generally within ±2.5 mm, indicating the volume of interest demands a uniform microstructure within 5 mm along length direction. The Gleeble® specimen geometry was accordingly designed through finite element modeling to control the thermal gradients throughout the volume of interest. Gas quench cooling was used to replicate fast cooling curves from welding. This methodology was validated by thermal gradients measurement during Gleeble® experiments and microstructural characterizations. 

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