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15 November 2022, Volume 40 Issue 6
Previous Issue
BOUNDARY INTEGRAL EQUATIONS FOR ISOTROPIC LINEAR ELASTICITY
Benjamin Stamm, Shuyang Xiang
2022, 40(6): 835864. DOI:
10.4208/jcm.2103m20190031
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This articles first investigates boundary integral operators for the threedimensional isotropic linear elasticity of a biphasic model with piecewise constant Lamé coefficients in the form of a bounded domain of arbitrary shape surrounded by a background material. In the simple case of a spherical inclusion, the vector spherical harmonics consist of eigenfunctions of the single and double layer boundary operators and we provide their spectra. Further, in the case of many spherical inclusions with isotropic materials, each with its own set of Lamé parameters, we propose an integral equation and a subsequent Galerkin discretization using the vector spherical harmonics and apply the discretization to several numerical test cases.
STABILIZED NONCONFORMING MIXED FINITE ELEMENT METHOD FOR LINEAR ELASTICITY ON RECTANGULAR OR CUBIC MESHES
Bei Zhang, Jikun Zhao, Minghao Li, Hongru Chen
2022, 40(6): 865881. DOI:
10.4208/jcm.2103m20200143
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Based on the primal mixed variational formulation, a stabilized nonconforming mixed finite element method is proposed for the linear elasticity on rectangular and cubic meshes. Two kinds of penalty terms are introduced in the stabilized mixed formulation, which are the jump penalty term for the displacement and the divergence penalty term for the stress. We use the classical nonconforming rectangular and cubic elements for the displacement and the discontinuous piecewise polynomial space for the stress, where the discrete space for stress are carefully chosen to guarantee the wellposedness of discrete formulation. The stabilized mixed method is lockingfree. The optimal convergence order is derived in the
L
^{2}
norm for stress and in the broken
H
^{1}
norm and
L
^{2}
norm for displacement. A numerical test is carried out to verify the optimal convergence of the stabilized method.
AN
L
^{∞}
SECOND ORDER CARTESIAN METHOD FOR 3D ANISOTROPIC INTERFACE PROBLEMS
Baiying Dong, Xiufeng Feng, Zhilin Li
2022, 40(6): 882912. DOI:
10.4208/jcm.2103m20200107
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A second order accurate method in the infinity norm is proposed for general three dimensional anisotropic elliptic interface problems in which the solution and its derivatives, the coefficients, and source terms all can have finite jumps across one or several arbitrary smooth interfaces. The method is based on the 2D finite elementfinite difference (FEFD) method but with substantial differences in method derivation, implementation, and convergence analysis. One of challenges is to derive 3D interface relations since there is no invariance anymore under coordinate system transforms for the partial differential equations and the jump conditions. A finite element discretization whose coefficient matrix is a symmetric semipositive definite is used away from the interface; and the maximum preserving finite difference discretization whose coefficient matrix part is an Mmatrix is constructed at irregular elements where the interface cuts through. We aim to get a sharp interface method that can have second order accuracy in the pointwise norm. We show the convergence analysis by splitting errors into several parts. Nontrivial numerical examples are presented to confirm the convergence analysis.
IMPROVED HARMONIC INCOMPATIBILITY REMOVAL FOR SUSCEPTIBILITY MAPPING VIA REDUCTION OF BASIS MISMATCH
Chenglong Bao, Jianfeng Cai, Jae Kyu Choi, Bin Dong, Ke Wei
2022, 40(6): 913935. DOI:
10.4208/jcm.2103m20190256
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In quantitative susceptibility mapping (QSM), the background field removal is an essential data acquisition step because it has a significant effect on the restoration quality by generating a harmonic incompatibility in the measured local field data. Even though the sparsity based first generation harmonic incompatibility removal (1GHIRE) model has achieved the performance gain over the traditional approaches, the 1GHIRE model has to be further improved as there is a basis mismatch underlying in numerically solving Poisson’s equation for the background removal. In this paper, we propose the second generation harmonic incompatibility removal (2GHIRE) model to reduce a basis mismatch, inspired by the balanced approach in the tight frame based image restoration. Experimental results shows the superiority of the proposed 2GHIRE model both in the restoration qualities and the computational efficiency.
A TWOGRID FINITE ELEMENT APPROXIMATION FOR NONLINEAR TIME FRACTIONAL TWOTERM MIXED SUBDIFFUSION AND DIFFUSION WAVE EQUATIONS
Yanping Chen, Qiling Gu, Qingfeng Li, Yunqing Huang
2022, 40(6): 936954. DOI:
10.4208/jcm.2104m20200332
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In this paper, we develop a twogrid method (TGM) based on the FEM for 2D nonlinear time fractional twoterm mixed subdiffusion and diffusion wave equations. A twogrid algorithm is proposed for solving the nonlinear system, which consists of two steps: a nonlinear FE system is solved on a coarse grid, then the linearized FE system is solved on the fine grid by Newton iteration based on the coarse solution. The fully discrete numerical approximation is analyzed, where the Galerkin finite element method for the space derivatives and the finite difference scheme for the time Caputo derivative with order
α
∈ (1, 2) and
α
_{1}
∈ (0, 1). Numerical stability and optimal error estimate
O
(
h
^{r+1}
+
H
^{2r+2}
+
τ
^{min3α,2α1}}
) in
L
^{2}
norm are presented for twogrid scheme, where t, H and h are the time step size, coarse grid mesh size and fine grid mesh size, respectively. Finally, numerical experiments are provided to confirm our theoretical results and effectiveness of the proposed algorithm.
A STOCHASTIC ALGORITHM FOR FAULT INVERSE PROBLEMS IN ELASTIC HALF SPACE WITH PROOF OF CONVERGENCE
Darko Volkov
2022, 40(6): 955976. DOI:
10.4208/jcm.2104m20200262
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A general stochastic algorithm for solving mixed linear and nonlinear problems was introduced in [11]. We show in this paper how it can be used to solve the fault inverse problem, where a planar fault in elastic halfspace and a slip on that fault have to be reconstructed from noisy surface displacement measurements. With the parameter giving the plane containing the fault denoted by
m
and the regularization parameter for the linear part of the inverse problem denoted by
C
, both modeled as random variables, we derive a formula for the posterior marginal of
m
. Modeling
C
as a random variable allows to sweep through a wide range of possible values which was shown to be superior to selecting a fixed value [11]. We prove that this posterior marginal of
m
is convergent as the number of measurement points and the dimension of the space for discretizing slips increase. Simply put, our proof only assumes that the regularized discrete error functional for processing measurements relates to an order 1 quadrature rule and that the union of the finitedimensional spaces for discretizing slips is dense. Our proof relies on trace class operator theory to show that an adequate sequence of determinants is uniformly bounded. We also explain how our proof can be extended to a whole class of inverse problems, as long as some basic requirements are met. Finally, we show numerical simulations that illustrate the numerical convergence of our algorithm.
STABLE BOUNDARY CONDITIONS AND DISCRETIZATION FOR
P
_{N}
EQUATIONS
Jonas Bünger, Neeraj Sarna, Manuel Torrilhon
2022, 40(6): 9771003. DOI:
10.4208/jcm.2104m20190231
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A solution to the linear Boltzmann equation satisfies an energy bound, which reflects a natural fact: The energy of particles in a finite volume is bounded in time by the energy of particles initially occupying the volume augmented by the energy transported into the volume by particles entering the volume over time. In this paper, we present boundary conditions (BCs) for the spherical harmonic (
P
_{N}
) approximation, which ensure that this fundamental energy bound is satisfied by the
P
_{N}
approximation. Our BCs are compatible with the characteristic waves of
P
_{N}
equations and determine the incoming waves uniquely. Both, energy bound and compatibility, are shown on abstract formulations of
P
_{N}
equations and BCs to isolate the necessary structures and properties. The BCs are derived from a Marshak type formulation of BC and base on a nonclassical even/oddclassification of spherical harmonic functions and a stabilization step, which is similar to the truncation of the series expansion in the
P
_{N}
method. We show that summation by parts (SBP) finite differences on staggered grids in space and the method of simultaneous approximation terms (SAT) allows to maintain the energy bound also on the semidiscrete level.
A STOCHASTIC TRUSTREGION FRAMEWORK FOR POLICY OPTIMIZATION
Mingming Zhao, Yongfeng Li, Zaiwen Wen
2022, 40(6): 10041030. DOI:
10.4208/jcm.2104m20210007
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In this paper, we study a few challenging theoretical and numerical issues on the well known trust region policy optimization for deep reinforcement learning. The goal is to find a policy that maximizes the total expected reward when the agent acts according to the policy. The trust region subproblem is constructed with a surrogate function coherent to the total expected reward and a general distance constraint around the latest policy. We solve the subproblem using a preconditioned stochastic gradient method with a line search scheme to ensure that each step promotes the model function and stays in the trust region. To overcome the bias caused by sampling to the function estimations under the random settings, we add the empirical standard deviation of the total expected reward to the predicted increase in a ratio in order to update the trust region radius and decide whether the trial point is accepted. Moreover, for a Gaussian policy which is commonly used for continuous action space, the maximization with respect to the mean and covariance is performed separately to control the entropy loss. Our theoretical analysis shows that the deterministic version of the proposed algorithm tends to generate a monotonic improvement of the total expected reward and the global convergence is guaranteed under moderate assumptions. Comparisons with the stateoftheart methods demonstrate the effectiveness and robustness of our method over robotic controls and game playings from OpenAI Gym.
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