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A characteristics-mixed finite element method for Burgers' equation
Huanzhen Chen,Ziwen Jiang 한국전산응용수학회 2004 Journal of applied mathematics & informatics Vol.15 No.-
In this paper, we propose a new mixed nite element method,called the characteristics-mixed method, for approximating the solution toBurgers’ equation. This method is based upon a space-time variationalform of Burgers’ equation. The hyperbolic part of the equation is approx-imated along the characteristics in time and the diusion part is approx-imated by a mixed nite element method of lowest order. The schemeis locally conservative since uid is transported along the approximatecharacteristics on the discrete level and the test function can be piece-wise constant. Our analysis show the new method approximate the scalarunknown and the vector ux optimally and simultaneously. We also showthis scheme has much smaller time-truncation errors than those of standardmethods. Numerical example is presented to show that the new scheme iseasily implemented, shocks and boundary layers are handled with almostno oscillations.One of the contributions of the paper is to show how the optimal errorestimates in L2(Ωare obtained which are much more dicult than inthe standard nite element methods. These results seem to be new in theliterature of nite element methods.
A CHARACTERISTICS-MIXED FINITE ELEMENT METHOD FOR BURGERS' EQUATION
Chen, Huanzhen,Jiang, Ziwen 한국전산응용수학회 2004 Journal of applied mathematics & informatics Vol.15 No.1
In this paper, we propose a new mixed finite element method, called the characteristics-mixed method, for approximating the solution to Burgers' equation. This method is based upon a space-time variational form of Burgers' equation. The hyperbolic part of the equation is approximated along the characteristics in time and the diffusion part is approximated by a mixed finite element method of lowest order. The scheme is locally conservative since fluid is transported along the approximate characteristics on the discrete level and the test function can be piecewise constant. Our analysis show the new method approximate the scalar unknown and the vector flux optimally and simultaneously. We also show this scheme has much smaller time-truncation errors than those of standard methods. Numerical example is presented to show that the new scheme is easily implemented, shocks and boundary layers are handled with almost no oscillations. One of the contributions of the paper is to show how the optimal error estimates in $L^2(\Omega)$ are obtained which are much more difficult than in the standard finite element methods. These results seem to be new in the literature of finite element methods.
Jiang, Ziwen,Chen, Huanzhen 한국전산응용수학회 1998 Journal of applied mathematics & informatics Vol.5 No.1
Mixed finite element method is developed to approxi-mate the solution of the initial-boundary value problem for a strongly nonlinear second-order hyperbolic equation in divergence form. Exis-tence and uniqueness of the approximation are proved and optimal-order $L\infty$-in-time $L^2$-in-space a priori error estimates are derived for both the scalar and vector functions approximated by the method.
Liyun Yang,Fei Zhang,Huanzhen Xie,Siyu Chen,Changyu Lin 대한토목학회 2022 KSCE JOURNAL OF CIVIL ENGINEERING Vol.26 No.3
To investigate the dynamic mechanical properties and fracturing behavior of the red sandrock-polypropylene fiber reinforced concrete (R-PPFRC) composite, R-PPFRC composite specimens with different Polypropylene fiber (PPF) dosage were fabricated. Conducted quasi-static compression experiments and dynamic experiments by using Split Hopkinson pressure bar (SHPB) apparatus and testing machine. The fracture characteristics and process of composite specimens were analyzed with the assistance of ultra-high-velocity camera photography. It can be concluded that the dynamic increase factor and the dynamic compressive strength of composite specimens R-PPFRC increase with the increase of load level, indicating a remarkably strain rate strengthening effect. The optimal PPF content is determined finally. The dynamic impact toughness, ultimate toughness, and dissipation energy of R-PPFRC are significantly affected by the load level and the PPF content. The fracturing process of the composite specimen is divided into four phases, crack initiation, crack expansion, crack cluster and ultimate failure. After going through these phases, the ultimate failure pattern is mainly shear and split mixed failure.