This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and tw...This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and two scenarios for thermal distribution.The material properties are assessed under two conditions,i.e.,temperature dependence and temperature independence.The theoretical framework for the beams is based on the higher-order shear deformation theory,which incorporates shear deformations with higher-order polynomials.The governing equations are established from the Lagrange equations,and the beam displacement fields are approximated by the Chebyshev polynomials.Numerical simulations are performed to evaluate the effects of thermal load,slenderness,boundary condition(BC),and porosity distribution on the buckling and vibration behaviors of metal foam beams.The findings highlight the significant influence of temperature-dependent(TD)material properties on metal foam beams'buckling and vibration responses.展开更多
Recent years have witnessed growing interests in solving partial differential equations by deep neural networks,especially in the high-dimensional case.Unlike classical numerical methods,such as finite difference meth...Recent years have witnessed growing interests in solving partial differential equations by deep neural networks,especially in the high-dimensional case.Unlike classical numerical methods,such as finite difference method and finite element method,the enforcement of boundary conditions in deep neural networks is highly nontrivial.One general strategy is to use the penalty method.In the work,we conduct a comparison study for elliptic problems with four different boundary conditions,i.e.,Dirichlet,Neumann,Robin,and periodic boundary conditions,using two representative methods:deep Galerkin method and deep Ritz method.In the former,the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter.Therefore,it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions.However,by a number of examples,we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions.Besides,in some cases,when the boundary condition can be implemented in an exact manner,we find that such a strategy not only provides a better approximate solution but also facilitates the training process.展开更多
In this paper, the jacket platform is simulated by a non-uniform cantilever beam subjected to axial force. Based on the Hamilton theory, the equation of bending motion is developed and solved by the classical Ritz met...In this paper, the jacket platform is simulated by a non-uniform cantilever beam subjected to axial force. Based on the Hamilton theory, the equation of bending motion is developed and solved by the classical Ritz method combined with the pseudo-excitation method for random responses with non-classical damping. Usually, random responses of this continuous structure are obtained by orthogonality of modes, and some normal modes of the structure are needed, causing inconvenience for the analysis of the non-uniform beam whose normal modes are not easy to be obtained. However, if the pseudo-excitation method is extended to calculate random responses by combining it with the classical Ritz method, the responses of a non-uniform beam, such as auto-PSD function, cross-PSD and higher spectral moments, can be solved directly avoiding the calculation of normal modes. The numerical results show that the present method is effective and useful in aseismic design of platforms.展开更多
The stress field in granular soils heap(including piled coal) will have a non-negligible impact on the settlement of the underlying soils. It is usually obtained by measurements and numerical simulations.Because the f...The stress field in granular soils heap(including piled coal) will have a non-negligible impact on the settlement of the underlying soils. It is usually obtained by measurements and numerical simulations.Because the former method is not reliable as pressure cells instrumented on the interface between piled coal and the underlying soft soil do not work well, results from numerical methods alone are necessary to be doubly checked with one more method before they are extended to more complex cases. The generalized stress field in granular soils heap is analyzed with Rayleighe Ritz method. The problem is divided into two cases: case A without horizontal constraint on the base and case B with horizontal constraint on the base. In both cases, the displacement functions u(x, y) and v(x, y) are assumed to be cubic polynomials with 12 undetermined parameters, which will satisfy the Cauchy’s partial differential equations, generalized Hooke’s law and boundary equations. A function is built with the Rayleighe Ritz method according to the principle of minimum potential energy, and the problem is converted into solving two undetermined parameters through the variation of the function, while the other parameters are expressed in terms of these two parameters. By comparison of results from the Rayleighe Ritz method and numerical simulations, it is demonstrated that the Rayleighe Ritz method is feasible to study the generalized stress field in granular soils heap. Solutions from numerical methods are verified before being extended to more complicated cases.展开更多
In this paper,we analyse the equal width（EW） wave equation by using the mesh-free reproducing kernel particle Ritz（kp-Ritz） method.The mesh-free kernel particle estimate is employed to approximate the displacement...In this paper,we analyse the equal width（EW） wave equation by using the mesh-free reproducing kernel particle Ritz（kp-Ritz） method.The mesh-free kernel particle estimate is employed to approximate the displacement field.A system of discrete equations is obtained through the application of the Ritz minimization procedure to the energy expressions.The effectiveness of the kp-Ritz method for the EW wave equation is investigated by numerical examples in this paper.展开更多
A rotor manipulation mechanism for micro unmanned helicopter utilizing the inertia and the elasticity of the rotor is introduced. The lagging motion equation of the rotor blades is established, and then the natural fr...A rotor manipulation mechanism for micro unmanned helicopter utilizing the inertia and the elasticity of the rotor is introduced. The lagging motion equation of the rotor blades is established, and then the natural frequencies and mode shapes of the blade for the helicopter are studied by using beam characteristic orthogonal polynomials by the Rayleigh-Ritz method. The variation of natural frequencies with the speed of rotation and the mode shapes at different rotational speeds are plotted. The using of orthogonal polynomials for the bending shapes enables the computation of higher natural frequencies of any order to be accomplished without facing any difficulties.展开更多
In this paper,we propose a method for solving semilinear elliptical equa-tions using a ResNet with ReLU2 activations.Firstly,we present a comprehensive formulation based on the penalized variational form of the ellipt...In this paper,we propose a method for solving semilinear elliptical equa-tions using a ResNet with ReLU2 activations.Firstly,we present a comprehensive formulation based on the penalized variational form of the elliptical equations.We then apply the Deep Ritz Method,which works for a wide range of equations.We obtain an upper bound on the errors between the acquired solutions and the true solutions in terms of the depth D,width W of the ReLU2 ResNet,and the num-ber of training samples n.Our simulation results demonstrate that our method can effectively overcome the curse of dimensionality and validate the theoretical results.展开更多
We propose a deep learning-based method,the Deep Ritz Method,for numerically solving variational problems,particularly the ones that arise from par-tial differential equations.The Deep Ritz Method is naturally nonline...We propose a deep learning-based method,the Deep Ritz Method,for numerically solving variational problems,particularly the ones that arise from par-tial differential equations.The Deep Ritz Method is naturally nonlinear,naturally adaptive and has the potential to work in rather high dimensions.The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning.We illustrate the method on several problems including some eigenvalue problems.展开更多
Using deep neural networks to solve PDEs has attracted a lot of attentions recently.However,why the deep learning method works is falling far behind its empirical success.In this paper,we provide a rigorous numerical ...Using deep neural networks to solve PDEs has attracted a lot of attentions recently.However,why the deep learning method works is falling far behind its empirical success.In this paper,we provide a rigorous numerical analysis on deep Ritz method(DRM)[47]for second order elliptic equations with Neumann boundary conditions.We establish the first nonasymptotic convergence rate in H^(1)norm for DRM using deep networks with ReLU^(2)activation functions.In addition to providing a theoretical justification of DRM,our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of number of training samples.Technically,we derive bound on the approximation error of deep ReLU^(2)network in C^(1)norm and bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm and ReLU^(2)network,both of which are of independent interest.展开更多
文摘This study presents the Chebyshev polynomials-based Ritz method to examine the thermal buckling and free vibration characteristics of metal foam beams.The analyses include three models for porosity distribution and two scenarios for thermal distribution.The material properties are assessed under two conditions,i.e.,temperature dependence and temperature independence.The theoretical framework for the beams is based on the higher-order shear deformation theory,which incorporates shear deformations with higher-order polynomials.The governing equations are established from the Lagrange equations,and the beam displacement fields are approximated by the Chebyshev polynomials.Numerical simulations are performed to evaluate the effects of thermal load,slenderness,boundary condition(BC),and porosity distribution on the buckling and vibration behaviors of metal foam beams.The findings highlight the significant influence of temperature-dependent(TD)material properties on metal foam beams'buckling and vibration responses.
基金the grants NSFC 11971021National Key R&D Program of China(No.2018YF645B0204404)NSFC 11501399(R.Du)。
文摘Recent years have witnessed growing interests in solving partial differential equations by deep neural networks,especially in the high-dimensional case.Unlike classical numerical methods,such as finite difference method and finite element method,the enforcement of boundary conditions in deep neural networks is highly nontrivial.One general strategy is to use the penalty method.In the work,we conduct a comparison study for elliptic problems with four different boundary conditions,i.e.,Dirichlet,Neumann,Robin,and periodic boundary conditions,using two representative methods:deep Galerkin method and deep Ritz method.In the former,the PDE residual is minimized in the least-squares sense while the corresponding variational problem is minimized in the latter.Therefore,it is reasonably expected that deep Galerkin method works better for smooth solutions while deep Ritz method works better for low-regularity solutions.However,by a number of examples,we observe that deep Ritz method can outperform deep Galerkin method with a clear dependence of dimensionality even for smooth solutions and deep Galerkin method can also outperform deep Ritz method for low-regularity solutions.Besides,in some cases,when the boundary condition can be implemented in an exact manner,we find that such a strategy not only provides a better approximate solution but also facilitates the training process.
文摘In this paper, the jacket platform is simulated by a non-uniform cantilever beam subjected to axial force. Based on the Hamilton theory, the equation of bending motion is developed and solved by the classical Ritz method combined with the pseudo-excitation method for random responses with non-classical damping. Usually, random responses of this continuous structure are obtained by orthogonality of modes, and some normal modes of the structure are needed, causing inconvenience for the analysis of the non-uniform beam whose normal modes are not easy to be obtained. However, if the pseudo-excitation method is extended to calculate random responses by combining it with the classical Ritz method, the responses of a non-uniform beam, such as auto-PSD function, cross-PSD and higher spectral moments, can be solved directly avoiding the calculation of normal modes. The numerical results show that the present method is effective and useful in aseismic design of platforms.
文摘The stress field in granular soils heap(including piled coal) will have a non-negligible impact on the settlement of the underlying soils. It is usually obtained by measurements and numerical simulations.Because the former method is not reliable as pressure cells instrumented on the interface between piled coal and the underlying soft soil do not work well, results from numerical methods alone are necessary to be doubly checked with one more method before they are extended to more complex cases. The generalized stress field in granular soils heap is analyzed with Rayleighe Ritz method. The problem is divided into two cases: case A without horizontal constraint on the base and case B with horizontal constraint on the base. In both cases, the displacement functions u(x, y) and v(x, y) are assumed to be cubic polynomials with 12 undetermined parameters, which will satisfy the Cauchy’s partial differential equations, generalized Hooke’s law and boundary equations. A function is built with the Rayleighe Ritz method according to the principle of minimum potential energy, and the problem is converted into solving two undetermined parameters through the variation of the function, while the other parameters are expressed in terms of these two parameters. By comparison of results from the Rayleighe Ritz method and numerical simulations, it is demonstrated that the Rayleighe Ritz method is feasible to study the generalized stress field in granular soils heap. Solutions from numerical methods are verified before being extended to more complicated cases.
基金Project supported by the Natural Science Foundation of Zhejiang Province,China (Grant No. Y6110007)
文摘In this paper,we analyse the equal width（EW） wave equation by using the mesh-free reproducing kernel particle Ritz（kp-Ritz） method.The mesh-free kernel particle estimate is employed to approximate the displacement field.A system of discrete equations is obtained through the application of the Ritz minimization procedure to the energy expressions.The effectiveness of the kp-Ritz method for the EW wave equation is investigated by numerical examples in this paper.
基金This work was supported by the "985"foundation of China(No.082200102).
文摘A rotor manipulation mechanism for micro unmanned helicopter utilizing the inertia and the elasticity of the rotor is introduced. The lagging motion equation of the rotor blades is established, and then the natural frequencies and mode shapes of the blade for the helicopter are studied by using beam characteristic orthogonal polynomials by the Rayleigh-Ritz method. The variation of natural frequencies with the speed of rotation and the mode shapes at different rotational speeds are plotted. The using of orthogonal polynomials for the bending shapes enables the computation of higher natural frequencies of any order to be accomplished without facing any difficulties.
基金supported by the National Key Research and Development Program of China(Grant No.2020YFA0714200)the National Nature Science Foundation of China(Grant Nos.12125103,12071362,12371424,12371441)supported by the Fundamental Research Funds for the Central Universities.The numerical calculations have been done at the Supercomputing Center of Wuhan University.
文摘In this paper,we propose a method for solving semilinear elliptical equa-tions using a ResNet with ReLU2 activations.Firstly,we present a comprehensive formulation based on the penalized variational form of the elliptical equations.We then apply the Deep Ritz Method,which works for a wide range of equations.We obtain an upper bound on the errors between the acquired solutions and the true solutions in terms of the depth D,width W of the ReLU2 ResNet,and the num-ber of training samples n.Our simulation results demonstrate that our method can effectively overcome the curse of dimensionality and validate the theoretical results.
基金supported in part by the National Key Basic Research Program of China 2015CB856000Major Program of NNSFC under Grant 91130005,DOE Grant DE-SC0009248ONR Grant N00014-13-1-0338.
文摘We propose a deep learning-based method,the Deep Ritz Method,for numerically solving variational problems,particularly the ones that arise from par-tial differential equations.The Deep Ritz Method is naturally nonlinear,naturally adaptive and has the potential to work in rather high dimensions.The framework is quite simple and fits well with the stochastic gradient descent method used in deep learning.We illustrate the method on several problems including some eigenvalue problems.
基金supported by the National Key Research and Development Program of China(No.2020YFA0714200)the Science and Technology Major Project of Hubei Province under Grant 2021AAA010+2 种基金the National Science Foundation of China(Nos.12125103,12071362,11871474,11871385)the Natural Science Foundation of Hubei Province(No.2019CFA007)by the research fund of KLATASDSMOE.
文摘Using deep neural networks to solve PDEs has attracted a lot of attentions recently.However,why the deep learning method works is falling far behind its empirical success.In this paper,we provide a rigorous numerical analysis on deep Ritz method(DRM)[47]for second order elliptic equations with Neumann boundary conditions.We establish the first nonasymptotic convergence rate in H^(1)norm for DRM using deep networks with ReLU^(2)activation functions.In addition to providing a theoretical justification of DRM,our study also shed light on how to set the hyperparameter of depth and width to achieve the desired convergence rate in terms of number of training samples.Technically,we derive bound on the approximation error of deep ReLU^(2)network in C^(1)norm and bound on the Rademacher complexity of the non-Lipschitz composition of gradient norm and ReLU^(2)network,both of which are of independent interest.
