scholarly journals Tensor-based multiscale method for diffusion problems in quasi-periodic heterogeneous media

2018 ◽  
Vol 52 (3) ◽  
pp. 869-891
Author(s):  
Quentin Ayoul-Guilmard ◽  
Anthony Nouy ◽  
Christophe Binetruy
Keyword(s):  
Numerical Simulation ◽  
Numerical Methods ◽  
Sobolev Space ◽  
Product Space ◽  
Heterogeneous Media ◽  
Low Rank ◽  
Weak Formulation ◽  
Tensor Product Space ◽  
Periodic Setting ◽  
Diffusion Problems

This paper proposes to address the issue of complexity reduction for the numerical simulation of multiscale media in a quasi-periodic setting. We consider a stationary elliptic diffusion equation defined on a domain D such that D̅ is the union of cells {D̅i}i∈I and we introduce a two-scale representation by identifying any function v(x) defined on D with a bi-variate function v(i,y), where i ∈ I relates to the index of the cell containing the point x and y ∈ Y relates to a local coordinate in a reference cell Y. We introduce a weak formulation of the problem in a broken Sobolev space V(D) using a discontinuous Galerkin framework. The problem is then interpreted as a tensor-structured equation by identifying V(D) with a tensor product space ℝI⊗ V(Y) of functions defined over the product set I × Y. Tensor numerical methods are then used in order to exploit approximability properties of quasi-periodic solutions by low-rank tensors.

DIRICHLET PROBLEM WITH STOCHASTIC COEFFICIENTS

2005 ◽  
Vol 05 (04) ◽  
pp. 555-568 ◽  
Author(s):  
INGO BULLA
Keyword(s):  
Sobolev Space ◽  
Existence And Uniqueness ◽  
Product Space ◽  
Space Variable ◽  
Dirichlet Condition ◽  
Elliptic Partial Differential Equations ◽  
Random Coefficients ◽  
Stochastic Variable ◽  
Random Data ◽  
Tensor Product Space

We consider general second-order linear elliptic partial differential equations having random coefficients and random data and fulfilling the homogeneous Dirichlet condition. We prove the existence and uniqueness of the weak solution in a certain tensor product space which is suitably completed to make it a Hilbert space. The factors of this space are a Sobolev space of functions depending on the space variable and a general Sobolev space of functions depending on the stochastic variable.

scholarly journals Fractal Frames of Functions on the Rectangle

2021 ◽  
Vol 5 (2) ◽  
pp. 42
Author(s):  
María A. Navascués ◽  
Ram Mohapatra ◽  
Md. Nasim Akhtar
Keyword(s):  
Tensor Product ◽  
Product Space ◽  
Integrable Function ◽  
Two Dimensional ◽  
Tensor Product Space ◽  
Fractal Functions

In this paper, we define fractal bases and fractal frames of L2(I×J), where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of L2(I×J) with the tensor product space L2(I)⨂L2(J). First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of L2(I×J) composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable.

Focus Control Aided by Numerical Simulation in Heterogeneous Media for High-Intensity Focused Ultrasound Treatment

2013 ◽  
Vol 52 (7S) ◽  
pp. 07HF01 ◽  
Author(s):  
Ryuta Narumi ◽  
Kosuke Matsuki ◽  
Shigeru Mitarai ◽  
Takashi Azuma ◽  
Kohei Okita ◽  
...  
Keyword(s):  
Numerical Simulation ◽  
High Intensity ◽  
Focused Ultrasound ◽  
Heterogeneous Media ◽  
High Intensity Focused Ultrasound ◽  
Ultrasound Treatment

scholarly journals Wave Propagation Across Acoustic/Biot’s Media: A Finite-Difference Method

2013 ◽  
Vol 13 (4) ◽  
pp. 985-1012 ◽  
Author(s):  
Guillaume Chiavassa ◽  
Bruno Lombard
Keyword(s):  
Wave Propagation ◽  
Numerical Methods ◽  
Heterogeneous Media ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Compressional Wave ◽  
Interface Conditions ◽  
Immersed Interface Method ◽  
Fluid Medium ◽  
Strang Splitting

AbstractNumerical methods are developed to simulate the wave propagation in heterogeneous 2D fluid/poroelastic media. Wave propagation is described by the usual acoustics equations (in the fluid medium) and by the low-frequency Biot’s equations (in the porous medium). Interface conditions are introduced to model various hydraulic contacts between the two media: open pores, sealed pores, and imperfect pores. Well-posedness of the initial-boundary value problem is proven. Cartesian grid numerical methods previously developed in porous heterogeneous media are adapted to the present context: a fourth-order ADER scheme with Strang splitting for time- marching; a space-time mesh-refinement to capture the slow compressional wave predicted by Biot’s theory; and an immersed interface method to discretize the interface conditions and to introduce a subcell resolution. Numerical experiments and comparisons with exact solutions are proposed for the three types of interface conditions, demonstrating the accuracy of the approach.

scholarly journals Numerical simulation of magma intrusion on the thermal evolution of low rank coal

2021 ◽  
Author(s):  
Jingyu Jiang ◽  
Ke Zhao ◽  
Yuanping Cheng ◽  
Shaojie Zheng ◽  
Shuo Zhang ◽  
...  
Keyword(s):  
Thermal Conductivity ◽  
Numerical Simulation ◽  
Coal Seam ◽  
Water Content ◽  
Thermal Evolution ◽  
Low Rank ◽  
Maximum Temperature ◽  
High Water Content ◽  
Low Rank Coal ◽  
Magma Intrusion

Abstract To study the effect of magma intrusion on the thermal evolution of low-rank coal with high water content, the mathematical relationship between water content variation and thermal conductivity of low-rank coal was analyzed by COMSOL Multiphysics numerical simulation and field validation. Taking Daxing Mine in Tiefa coalfield as the research background, the effects of magma finite time intrusion mechanism and water volatilization in coal on thermal evolution and organic maturity of coal seam are investigated in this paper. The results show that as the sill thickness increases, the thermal evolution temperature of the coal seam increases, the required thermal evolution time increases and the final retention temperature increases after the coal seam is cooled down. Approaching the magma, the maximum temperature that the coal seam can reach increases, the maximum temperature lasts longer, and the final temperature retained by the coal seam becomes higher. The increase of water content of coal makes the thermal conductivity increase, and the rate of heat transfer from coal seam is accelerated, and more heat is transferred to distant places in the same time. At the same time, the heat lost by the magma in the same time increases, the time required for the cooling of the magma decreases, and the maximum temperature reached by the underlying coal seam is significantly lower. The presence of moisture weakens the thermal evolution of the magma to the coal seam and reduces the expected maturity of the coal. The results of average random vitrinite reflectance (Ro) and moisture examination of coal samples collected at the Daxing Mine site verified the numerical simulation results of magma thermal evolution.

scholarly journals Numerical simulation of minor losses coefficient on the example of elbows

2018 ◽  
Vol 180 ◽  
pp. 02093
Author(s):  
Smyk Emil ◽  
Mrozik Dariusz ◽  
Olszewski Łukasz ◽  
Peszyński Kazimierz
Keyword(s):  
Numerical Simulation ◽  
Numerical Methods ◽  
Numerical Method ◽  
Cross Section ◽  
Analytical Methods ◽  
Circular Cross Section ◽  
Experimental Methods ◽  
Loss Coefficient ◽  
Complicated Problem ◽  
Minor Losses

Determining of minor losses coefficient is very complicated problem. Analytical methods are often very difficult and experimental methods are very expensive and time-consuming. Consequently, the use of numerical methods seems to be a good solution, but there are no publications describing this issue. Therefore, the paper is describing the numerical method of determining the minor loss coefficient ξ on the example of elbows with circular cross-section.

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