The linear constitutive equations and field equations of unsaturated soils were obtained through linearizing the nonlinear equations given in the first part of this work. The linear equations were expressed in the for...The linear constitutive equations and field equations of unsaturated soils were obtained through linearizing the nonlinear equations given in the first part of this work. The linear equations were expressed in the forms similar to Biot's equations for saturated porous media. The Darcy's laws of unsaturated soil were proved. It is shown that Biot's equations of saturated porous media are the simplification of the theory. All these illustrate that constructing constitutive relation of unsaturated soil on the base of mixture theory is rational.展开更多
The overall objectives to support analytically the mathematical background of hydraulics, linking the Navier-Stokes with hydraulic formulas, which origin is experimental but have wide and varied application. This, lea...The overall objectives to support analytically the mathematical background of hydraulics, linking the Navier-Stokes with hydraulic formulas, which origin is experimental but have wide and varied application. This, leads us study the inverse problem of the coefficients of differential equations, such as equations of the porous medium, Saint-Venant, and Reynolds, and accordingly with the order of derivatives. The research led us to see that the classic version suffers from a parameter that reflects the fractal and non-local character of the viscous interaction. Motivated by the concept of spatial occupancy rate, the authors set forth Navier-Stokes＇s fractional equation and the authors obtain the fractional Saint-Venant. In particular, the hydraulic gradient, or friction, is conceived as a fractional derivative of velocity. The friction factor is described as a linear operator acting on speed, so that the information it contains is transferred to the order of the derivative, so that the same is linearly related to the exponent of the friction factor. It states Darcy＇s non-linear law. The authors take a previous result that describes the nonlinear flow law with a leading term that contains a hyper-geometric function, whose parameters depend on the exponent of the friction factor and the exponent of the hydraulic radius. It searches the various laws of flow according to the best known laws of hydraulic resistance, such as Chezy and Manning.展开更多
We present an efficient deep learning method called coupled deep neural networks(CDNNs) for coupling of the Stokes and Darcy–Forchheimer problems. Our method compiles the interface conditions of the coupled problems ...We present an efficient deep learning method called coupled deep neural networks(CDNNs) for coupling of the Stokes and Darcy–Forchheimer problems. Our method compiles the interface conditions of the coupled problems into the networks properly and can be served as an efficient alternative to the complex coupled problems. To impose energy conservation constraints, the CDNNs utilize simple fully connected layers and a custom loss function to perform the model training process as well as the physical property of the exact solution. The approach can be beneficial for the following reasons: Firstly, we sample randomly and only input spatial coordinates without being restricted by the nature of samples.Secondly, our method is meshfree, which makes it more efficient than the traditional methods. Finally, the method is parallel and can solve multiple variables independently at the same time. We present the theoretical results to guarantee the convergence of the loss function and the convergence of the neural networks to the exact solution. Some numerical experiments are performed and discussed to demonstrate performance of the proposed method.展开更多
Forchheimer方程作为非达西渗流中广泛应用的基本方程之一,方程中A、B系数的确定一直是孔隙介质渗流领域中的热点及难点,不同学者根据渗流试验结果提出了不同的Forchheimer方程A、B系数的经验公式,但对于均质以及混合粒径的非均质条件...Forchheimer方程作为非达西渗流中广泛应用的基本方程之一,方程中A、B系数的确定一直是孔隙介质渗流领域中的热点及难点,不同学者根据渗流试验结果提出了不同的Forchheimer方程A、B系数的经验公式,但对于均质以及混合粒径的非均质条件下评价各经验公式适用性的研究较少。因此在渗流阻力试验的基础上,采用归一化目标函数和线性回归法评价了Forchheimer方程经验公式的适用性,为不同孔隙介质条件下Forchheimer方程经验公式的选取提供参考。结果表明:对于均质孔隙介质,Sidiropoulou公式对水力梯度有着很好的预测效果;对于2种混合粒径孔隙介质,在使用平均粒径的基础上,还应考虑混合粒径的质量比和大小因素,Macdonald公式的预测效果受混合粒径的质量比和大小影响较小,Kadlec and Knight公式对于水力梯度的预测结果较为稳定;对于5种混合粒径孔隙介质,使用d60作为特征粒径进行预测的效果较好,Kadlec and Knight公式对于系数A的预测效果较好,Ergun公式对于系数B的预测效果较好。研究结果能够为工程中均质及非均质松散砂砾石孔隙介质渗流计算的Forchheimer方程的选取提供依据。展开更多
Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/ Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal ...Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/ Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal error order for the porous media flow and a non-optimal error order for the fluid flow. In this article, we give a more rigorous of the error analysis for the fluid flow and obtain the optimal error estimates of the velocity and the pressure.展开更多
A new decoupled two-gird algorithm with the Newton iteration is proposed for solving the coupled Navier-Stokes/Darcy model which describes a fluid flow filtrating through porous media. Moreover the error estimate is g...A new decoupled two-gird algorithm with the Newton iteration is proposed for solving the coupled Navier-Stokes/Darcy model which describes a fluid flow filtrating through porous media. Moreover the error estimate is given, which shows that the same order of accuracy can be achieved as solving the system directly in the fine mesh when h = H2. Both theoretical analysis and numerical experiments illustrate the efficiency of the algorithm for solving the coupled problem.展开更多
文摘The linear constitutive equations and field equations of unsaturated soils were obtained through linearizing the nonlinear equations given in the first part of this work. The linear equations were expressed in the forms similar to Biot's equations for saturated porous media. The Darcy's laws of unsaturated soil were proved. It is shown that Biot's equations of saturated porous media are the simplification of the theory. All these illustrate that constructing constitutive relation of unsaturated soil on the base of mixture theory is rational.
文摘The overall objectives to support analytically the mathematical background of hydraulics, linking the Navier-Stokes with hydraulic formulas, which origin is experimental but have wide and varied application. This, leads us study the inverse problem of the coefficients of differential equations, such as equations of the porous medium, Saint-Venant, and Reynolds, and accordingly with the order of derivatives. The research led us to see that the classic version suffers from a parameter that reflects the fractal and non-local character of the viscous interaction. Motivated by the concept of spatial occupancy rate, the authors set forth Navier-Stokes＇s fractional equation and the authors obtain the fractional Saint-Venant. In particular, the hydraulic gradient, or friction, is conceived as a fractional derivative of velocity. The friction factor is described as a linear operator acting on speed, so that the information it contains is transferred to the order of the derivative, so that the same is linearly related to the exponent of the friction factor. It states Darcy＇s non-linear law. The authors take a previous result that describes the nonlinear flow law with a leading term that contains a hyper-geometric function, whose parameters depend on the exponent of the friction factor and the exponent of the hydraulic radius. It searches the various laws of flow according to the best known laws of hydraulic resistance, such as Chezy and Manning.
基金Project supported in part by the National Natural Science Foundation of China (Grant No.11771259)the Special Support Program to Develop Innovative Talents in the Region of Shaanxi Province+1 种基金the Innovation Team on Computationally Efficient Numerical Methods Based on New Energy Problems in Shaanxi Provincethe Innovative Team Project of Shaanxi Provincial Department of Education (Grant No.21JP013)。
文摘We present an efficient deep learning method called coupled deep neural networks(CDNNs) for coupling of the Stokes and Darcy–Forchheimer problems. Our method compiles the interface conditions of the coupled problems into the networks properly and can be served as an efficient alternative to the complex coupled problems. To impose energy conservation constraints, the CDNNs utilize simple fully connected layers and a custom loss function to perform the model training process as well as the physical property of the exact solution. The approach can be beneficial for the following reasons: Firstly, we sample randomly and only input spatial coordinates without being restricted by the nature of samples.Secondly, our method is meshfree, which makes it more efficient than the traditional methods. Finally, the method is parallel and can solve multiple variables independently at the same time. We present the theoretical results to guarantee the convergence of the loss function and the convergence of the neural networks to the exact solution. Some numerical experiments are performed and discussed to demonstrate performance of the proposed method.
文摘Forchheimer方程作为非达西渗流中广泛应用的基本方程之一,方程中A、B系数的确定一直是孔隙介质渗流领域中的热点及难点,不同学者根据渗流试验结果提出了不同的Forchheimer方程A、B系数的经验公式,但对于均质以及混合粒径的非均质条件下评价各经验公式适用性的研究较少。因此在渗流阻力试验的基础上,采用归一化目标函数和线性回归法评价了Forchheimer方程经验公式的适用性,为不同孔隙介质条件下Forchheimer方程经验公式的选取提供参考。结果表明:对于均质孔隙介质,Sidiropoulou公式对水力梯度有着很好的预测效果;对于2种混合粒径孔隙介质,在使用平均粒径的基础上,还应考虑混合粒径的质量比和大小因素,Macdonald公式的预测效果受混合粒径的质量比和大小影响较小,Kadlec and Knight公式对于水力梯度的预测结果较为稳定;对于5种混合粒径孔隙介质,使用d60作为特征粒径进行预测的效果较好,Kadlec and Knight公式对于系数A的预测效果较好,Ergun公式对于系数B的预测效果较好。研究结果能够为工程中均质及非均质松散砂砾石孔隙介质渗流计算的Forchheimer方程的选取提供依据。
基金Subsidized by NSFC(11571274 and 11171269)the Ph.D.Programs Foundation of Ministry of Education of China(20110201110027)
文摘Although the two-grid finite element decoupled scheme for mixed Navier-Stokes/ Darcy model in literatures has given the numerical results of optimal convergence order, the theoretical analysis only obtain the optimal error order for the porous media flow and a non-optimal error order for the fluid flow. In this article, we give a more rigorous of the error analysis for the fluid flow and obtain the optimal error estimates of the velocity and the pressure.
基金supported by National Foundation of Natural Science(11471092,11326231)Zhejiang Provincial Natural Science Foundation of China(LZ13A010003)
文摘A new decoupled two-gird algorithm with the Newton iteration is proposed for solving the coupled Navier-Stokes/Darcy model which describes a fluid flow filtrating through porous media. Moreover the error estimate is given, which shows that the same order of accuracy can be achieved as solving the system directly in the fine mesh when h = H2. Both theoretical analysis and numerical experiments illustrate the efficiency of the algorithm for solving the coupled problem.
