In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid.The implicit n...In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid.展开更多
On purpose of studying the sluggish diffusion of high-entropy alloys, three different face centered cubic Co-Cr-Cu-Fe-Ni high-entropy alloys were prepared, and assembled into three groups of sandwich- type diffusion m...On purpose of studying the sluggish diffusion of high-entropy alloys, three different face centered cubic Co-Cr-Cu-Fe-Ni high-entropy alloys were prepared, and assembled into three groups of sandwich- type diffusion multiple annealed at 1273, 1323, and 1373 K respectively. By means of the electron probe microanalyzer technique and recently developed numerical inverse method, the composition- dependent interdiffusivities at different temperatures were effectively evaluated by minimizing the residual between the model-predicted compositions/interdiffusion fluxes and the respectively experi- mental ones. After that, the tracer diffusivities were predicted based on the assessed mobility parameters and thermodynamic descriptions with the simplified ideal solution model. The comprehensive compari- son between the interdiffusivities/tracer diffusivities in the Co-Cr-Cu-Fe-Ni high-entropy alloys and those in sub-binary, ternary, quaternary and other quinary alloys indicates that the sluggish diffusion exists in interdiffusion instead of tracer diffusion for the present Co-Cr-Cu-Fe-Ni high-entropy alloys.展开更多
Study on the diffusion growth of ternary intermetallic compounds in Mg-Al-Zn based light-weight alloys is important due to its close interrelation with alloy property.However,there is a very lack of existing data due ...Study on the diffusion growth of ternary intermetallic compounds in Mg-Al-Zn based light-weight alloys is important due to its close interrelation with alloy property.However,there is a very lack of existing data due to difficulties in both experimental and computational aspects.The current work aims at presenting the experimental observation on the diffusion growth behavior of Φ phase at 360℃ as well as calculating its composition-dependent interdiffu sion coefficients.We designed and succes s fully fabricated four Mg-τ ternary diffusion couples annealed at 360℃ for different times,where the diffusion path goes across the Φ phase region and the diffusion growth of ternary intermetallic compound can be solely detected.In-situ observation of the time-dependent growth of Φ phase was performed to accurately determine the parabolic growth constant.The experimental data were then subjected to a numerical inverse method to generate a set of self-consistent interdiffusivities of the ternary intermetallic compounds,which can reproduce the presently observed diffusion growth behavior of Φ ternary intermetallic compound in Mg-τ diffusion couples.展开更多
基金supported by the National Natural Science Foundation of China(Grants 11472161,11102102,and 91130017)the Independent Innovation Foundation of Shandong University(Grant 2013ZRYQ002)the Natural Science Foundation of Shandong Province(Grant ZR2014AQ015)
文摘In this paper,we propose a numerical method to estimate the unknown order of a Riemann-Liouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid.The implicit numerical method is employed to solve the direct problem.For the inverse problem,we first obtain the fractional sensitivity equation by means of the digamma function,and then we propose an efficient numerical method,that is,the Levenberg-Marquardt algorithm based on a fractional derivative,to estimate the unknown order of a Riemann-Liouville fractional derivative.In order to demonstrate the effectiveness of the proposed numerical method,two cases in which the measurement values contain random measurement error or not are considered.The computational results demonstrate that the proposed numerical method could efficiently obtain the optimal estimation of the unknown order of a RiemannLiouville fractional derivative for a fractional Stokes' first problem for a heated generalized second grade fluid.
基金financially supported by the National Natural Science Foundation of China(Grant No.51474239)the National Key Research and Development Program of China(Grant No.2016YFB0301101)+1 种基金financial support from the Huxiang Youth Talent Plan released by Hunan Province,Chinathe project supported by State Key Laboratory of Powder Metallurgy Foundation,Central South University,Changsha,China
文摘On purpose of studying the sluggish diffusion of high-entropy alloys, three different face centered cubic Co-Cr-Cu-Fe-Ni high-entropy alloys were prepared, and assembled into three groups of sandwich- type diffusion multiple annealed at 1273, 1323, and 1373 K respectively. By means of the electron probe microanalyzer technique and recently developed numerical inverse method, the composition- dependent interdiffusivities at different temperatures were effectively evaluated by minimizing the residual between the model-predicted compositions/interdiffusion fluxes and the respectively experi- mental ones. After that, the tracer diffusivities were predicted based on the assessed mobility parameters and thermodynamic descriptions with the simplified ideal solution model. The comprehensive compari- son between the interdiffusivities/tracer diffusivities in the Co-Cr-Cu-Fe-Ni high-entropy alloys and those in sub-binary, ternary, quaternary and other quinary alloys indicates that the sluggish diffusion exists in interdiffusion instead of tracer diffusion for the present Co-Cr-Cu-Fe-Ni high-entropy alloys.
基金supported financially by the National Key Research and Development Program of China(No.2016YFB0701202)the National Natural Science Foundation of China(Nos.51801116 and 51901117)+1 种基金the Natural Science Foundation of Shandong Province(No.ZR2017BEM022)the Youth Fund of Shandong Academy of Sciences(Nos.2018QN0032 and 2019QN0023)。
文摘Study on the diffusion growth of ternary intermetallic compounds in Mg-Al-Zn based light-weight alloys is important due to its close interrelation with alloy property.However,there is a very lack of existing data due to difficulties in both experimental and computational aspects.The current work aims at presenting the experimental observation on the diffusion growth behavior of Φ phase at 360℃ as well as calculating its composition-dependent interdiffu sion coefficients.We designed and succes s fully fabricated four Mg-τ ternary diffusion couples annealed at 360℃ for different times,where the diffusion path goes across the Φ phase region and the diffusion growth of ternary intermetallic compound can be solely detected.In-situ observation of the time-dependent growth of Φ phase was performed to accurately determine the parabolic growth constant.The experimental data were then subjected to a numerical inverse method to generate a set of self-consistent interdiffusivities of the ternary intermetallic compounds,which can reproduce the presently observed diffusion growth behavior of Φ ternary intermetallic compound in Mg-τ diffusion couples.