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1. Effective floating volume: a highly parallelizable mesh-free approach for solving transient multiphysics problems in multi-scale geometries with non-linear material properties

   https://doi.org/10.1007/s00466-019-01797-x  

   Bahadori, Reza; Gutierrez, Hector (November 2019, Computational mechanics)

   A novel numerical method is proposed for the solution of transient multi-physics problems involving heat conduction, electrical current sharing and Joule heating. The innovation consists of a mesh-free Monte Carlo approach that eliminates or drastically reduces the particle scattering requirements typical of conventional Monte-Carlo methods. The proposed algorithm encapsulates a volume around each point that affects the solution at a given point in the domain; the volume includes other points that represent small perturbations along the path of energy transfer. The proposed method is highly parallelizable and amenable for GPU computing, and its computational performance was substantially increased by the elimination of scattered interpolation. The accuracy and simulation time of the proposed method are compared against a finite element solution and also against experimental results from existing literature. The proposed method provides accuracy comparable to that of finite element methods, achieving an order of magnitude reduction in simulation time. 

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2. An efficient positive‐definite block‐preconditioned finite volume solver for two‐sided fractional diffusion equations on composite mesh

   https://doi.org/10.1002/nla.2372  

   Dai, Pingfei; Jia, Jinhong; Wang, Hong; Wu, Qingbiao; Zheng, Xiangcheng (March 2021, Numerical Linear Algebra with Applications)

   Abstract It is known that the solutions to space‐fractional diffusion equations exhibit singularities near the boundary. Therefore, numerical methods discretized on the composite mesh, in which the mesh size is refined near the boundary, provide more precise approximations to the solutions. However, the coefficient matrices of the corresponding linear systems usually lose the diagonal dominance and are ill‐conditioned, which in turn affect the convergence behavior of the iteration methods.In this work we study a finite volume method for two‐sided fractional diffusion equations, in which a locally refined composite mesh is applied to capture the boundary singularities of the solutions. The diagonal blocks of the resulting three‐by‐three block linear system are proved to be positive‐definite, based on which we propose an efficient block Gauss–Seidel method by decomposing the whole system into three subsystems with those diagonal blocks as the coefficient matrices. To further accelerate the convergence speed of the iteration, we use T. Chan's circulant preconditioner³¹as the corresponding preconditioners and analyze the preconditioned matrices' spectra. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method and its strong potential in dealing with ill‐conditioned problems. While we have not proved the convergence of the method in theory, the numerical experiments show that the proposed method is convergent. 

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3. A 3D OpenFOAM based finite volume solver for incompressible Oldroyd-B model with infinity relaxation time

   Tian, Jing; Cai, Mingchao; Luo, Zhendong; You, Yuncheng; Chen, Goong (November 2019, Communications in Nonlinear Science Numerical Simulation)

   In this paper, we develop a Finite Volume solver for a 3D incompressible Oldroyd-B model with infinity relaxation time. The Finite Volume solver is implemented by using a lead- ing open-source computational mechanics software OpenFOAM. We have imposed the di- vergence free condition as a constraint on the momentum equation to derive a pressure equation and a predictor-corrector procedure is applied when solving the velocity field. Both stability analysis and numerical experiments are given to show the robustness and accuracy of our algorithm. Two concrete examples on a cubical domain and a dumbbell are computed and illustrated. 

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4. Numerical Model for Thaw Consolidation of Ice-Rich Permafrost Using the Finite Volume Approach

   https://doi.org/10.1061/9780784485989.017  

   Wang, Ziyi; Xiao, Ming (February 2025, American Society of Civil Engineers)

   This paper presents an implicit numerical model for one-dimensional thaw consolidation of saturated permafrost using finite volume approach. The model couples heat transfer with consolidation deformation and accounts for conduction, advection, phase change in heat transfer, and large strain in consolidation. The Crank–Nicolson method is used to obtain transient solutions. The overall approximation of the numerical scheme is of second-order accuracy. Numerical simulations are conducted to analyze the thaw consolidation behaviors in a finite soil layer. Numerical results indicate that, in a finite soil layer, thaw penetration and settlement have nonlinear relationships with the square root of time with decreasing rate. The excess pore water pressure and void ratio at the thaw front decrease with time. Thaw consolidation behaviors can be strongly influenced by the thermal conductivity of soil grains. The advection heat-transfer mechanism has a negligible effect on thaw consolidation in low-permeability soil. 

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5. Statistical analysis for a penalized EM algorithm in high-dimensional mixture linear regression model

   Wang, N; Zhang, X; Mai, Q (July 2024, Journal of machine learning research)

   The expectation-maximization (EM) algorithm and its variants are widely used in statistics. In high-dimensional mixture linear regression, the model is assumed to be a finite mixture of linear regression and the number of predictors is much larger than the sample size. The standard EM algorithm, which attempts to find the maximum likelihood estimator, becomes infeasible for such model. We devise a group lasso penalized EM algorithm and study its statistical properties. Existing theoretical results of regularized EM algorithms often rely on dividing the sample into many independent batches and employing a fresh batch of sample in each iteration of the algorithm. Our algorithm and theoretical analysis do not require sample-splitting, and can be extended to multivariate response cases. The proposed methods also have encouraging performances in numerical studies. 

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