Article
Engineering, Multidisciplinary
Peter Hansbo, Mats G. Larson
Summary: In this paper, a nonconforming rotated bilinear tetrahedral element is applied to the Stokes problem in R-3, demonstrating stability in combination with a piecewise linear, continuous approximation of the pressure. This element provides an approximation similar to the well-known Taylor-Hood element but with fewer degrees of freedom, and fulfills Korn's inequality, ensuring stability even when the Stokes equations are written on stress form for use in free surface flow.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2022)
Article
Mathematics, Applied
Haifeng Ji, Feng Wang, Jinru Chen, Zhilin Li
Summary: In this paper, a significant discovery has been made regarding nonconforming immersed finite element (IFE) methods for solving elliptic interface problems. It has been shown that IFE methods without penalties may not converge optimally in the presence of non-zero tangential derivative of the exact solution and coefficient jump on the interface. To address this issue, a new parameter-free nonconforming IFE method with additional terms on interface edges has been developed to recover the optimal convergence rates. The method has been shown to have unisolvent IFE basis functions on arbitrary triangles.
ESAIM-MATHEMATICAL MODELLING AND NUMERICAL ANALYSIS
(2023)
Article
Mathematics, Applied
Zhaoliang Meng, Jintao Cui, Zhongxuan Luo
Summary: A new nonparametric nonconforming quadrilateral finite element is introduced to approximate the general second-order elliptic problem in two dimensions, along with optimal numerical integration formulas. These formulas, derived on a reference quadrilateral and involving only two quadrature nodes, are not required to be exact for all shape functions and can be used with other low-order elements in numerical tests.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Liuchao Xiao, Meng Zhou, Jikun Zhao
Summary: The paper develops the nonconforming virtual element method (VEM) for the semilinear elliptic problem and approximates the nonlinear right-hand side using the L(2) projection. It proves the optimal convergence of the nonconforming VEM in the broken H-1 norm and carries out numerical experiments to support the theoretical results.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Computer Science, Software Engineering
Xiaosong Zhu, Youyuan Wang
Summary: In this paper, a new fully automatic pure hexahedron mesh generation technique is presented for an arbitrary 2D/3D geometric domain. Additionally, a polyhedral smoothed finite element method is proposed for efficient numerical simulation. Several demonstration examples are provided to verify the accuracy of the proposed methods.
COMPUTER-AIDED DESIGN
(2022)
Article
Mathematics, Applied
Do Y. Kwak, Hyeokjoo Park
Summary: In this work, a formal construction of a pointwise divergence-free basis in the nonconforming virtual element method for the Stokes problem is developed, extending the concept from the traditional finite element space. The elimination of the pressure variable from the mixed system and the achievement of a symmetric positive definite system are demonstrated through numerical tests, confirming the efficiency and accuracy of the proposed construction.
NUMERICAL ALGORITHMS
(2022)
Article
Computer Science, Software Engineering
Teseo Schneider, Yixin Hu, Xifeng Gao, Jeremie Dumas, Denis Zorin, Daniele Panozzo
Summary: The Finite Element Method (FEM) is commonly used to solve discrete Partial Differential Equations (PDEs) in engineering and graphics applications. This study introduces a set of benchmark problems and compares the performance of different element types for solving common elliptic PDEs using tetrahedral and hexahedral meshes. The results suggest that for certain problem sets and available mesh generation algorithms, quadratic tetrahedral elements perform well and can outperform hexahedral elements in structural analysis, thermal analysis, and low Reynolds number flows.
ACM TRANSACTIONS ON GRAPHICS
(2022)
Article
Engineering, Multidisciplinary
N. Staili, M. Rhoudaf
Summary: The aim of this paper is to simulate the two-dimensional stationary Stokes problem. The Stokes problem is reduced to a biharmonic one using the vorticity-Stream function formulation. The paper develops an approach to discretize the Laplace operator by the nonconforming P1 finite element. The convergence of the method is shown with the techniques of compactness, and an error estimate is proved for solutions in C-4 ((Ω) over bar). Numerical experiments are performed for the steady-driven cavity problem.
INTERNATIONAL JOURNAL OF COMPUTATIONAL METHODS
(2022)
Article
Mathematics, Applied
Jaeryun Yim, Dongwoo Sheen
Summary: The P1 nonconforming quadrilateral finite element space with periodic boundary conditions is studied. The dimension and basis of the space are determined using the concept of minimally essential discrete boundary conditions. The situation is found to be different based on the parity of the number of discretizations on coordinates. Several numerical schemes are proposed for solving elliptic problems with periodic boundary conditions, some of which involve solving linear equations with non-invertible matrices. The existence of corresponding numerical solutions is guaranteed using the Drazin inverse. The theoretical relationship between the numerical solutions is derived and confirmed by numerical results. Finally, the extension to three dimensions is provided.
NUMERICAL METHODS FOR PARTIAL DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
S. Bertoluzza, G. Manzini, M. Pennacchio, D. Prada
Summary: In this paper, we address the issue of designing robust stabilization terms for the nonconforming virtual element method. We transfer the problem of defining the stabilizing bilinear form to the dual space, which allows us to construct different bilinear forms with optimal or quasi-optimal stability bounds and error estimates. This approach relaxes the assumptions on the tessellation and proves optimality even under geometrical conditions that allow a mesh to have a very large number of arbitrarily small edges per element. We also numerically assess the performance of the VEM under different stabilizations using representative test cases.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Engineering, Mechanical
Subray R. Hegde, J. K. Rakshan Kumar, Pavankumar Sondar, Preetish C. Dsilva
Summary: The investigation revealed that numerous dents and nicks on the fan-blades were operating in a mild corrosive atmosphere. A pre-existing dent on a blade corroded to form a pre-crack, which eventually led to catastrophic failure due to mechanical imbalance of the fan-shaft caused by the blade failure. Grinding off surface defects to eliminate stress raisers is recommended to prevent such failures in the future.
ENGINEERING FAILURE ANALYSIS
(2021)
Article
Mathematics, Applied
Derrick Jones, Xu Zhang
Summary: This paper introduces a class of lowest-order nonconforming immersed finite element methods for solving two-dimensional Stokes interface problems, which do not require the solution mesh to align with the fluid interface and can use triangular or rectangular meshes. The new vector-valued IFE functions are constructed to approximate the interface jump conditions, and the approximation capabilities of these new IFE spaces for the Stokes interface problems are examined through numerical examples, showing optimal convergence rates.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Computer Science, Information Systems
Luca Quagliato, Seok Chang Ryu
Summary: A new FEA model was developed to investigate the design and tissue interaction of actively steerable needles with a rotational tip joint in soft tissue. The model allows simulation of needle insertion along non-predetermined paths and study of various steering motions by changing needle geometry and boundary conditions. The results demonstrate the importance of tip geometry in tissue damage gradient.
Article
Mathematics, Applied
E. P. Shurina, N. B. Itkina, N. Shtabel, E. Shtanko, A. Yu Kutishcheva, S. Markov, D. Dobrolubova
Summary: This paper considers the problem of calculating effective anisotropic thermal, electric and mechanical properties of composite media. It proposes numerical algorithms for homogenization based on upscaling technique and effective medium theory, utilizing mathematical modeling and multiscale non-conforming finite element methods. The study investigates the influence of physical properties and arrangement of microinclusions on effective tensors for different types of field excitation.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Shangyou Zhang
Summary: The paper proposes a stable and effective finite element method that satisfies both the discrete Korn inequality and the requirements of the Stokes problem. The linear conforming finite element is enriched by introducing some nonconforming bubbles.
JOURNAL OF NUMERICAL MATHEMATICS
(2023)
