A Stable Mixed Element Method for the Biharmonic Equation with First-Order Function Spaces

2017 ◽  
Vol 17 (4) ◽  
pp. 601-616 ◽  
Author(s):  
Zheng Li ◽  
Shuo Zhang
Keyword(s):  
Biharmonic Equation ◽  
Two Dimensions ◽  
Zero Order ◽  
Helmholtz Decomposition ◽  
Numerical Verification ◽  
Order Function ◽  
First Order ◽  
Mixed Element ◽  
New Formulation ◽  
Element Method

AbstractThis paper studies the mixed element method for the boundary value problem of the biharmonic equation {\Delta^{2}u=f} in two dimensions. We start from a {u\sim\nabla u\sim\nabla^{2}u\sim\operatorname{div}\nabla^{2}u} formulation that is discussed in [4] and construct its stability on {H^{1}_{0}(\Omega)\times\tilde{H}^{1}_{0}(\Omega)\times\bar{L}_{\mathrm{sym}}^% {2}(\Omega)\times H^{-1}(\operatorname{div},\Omega)}. Then we utilise the Helmholtz decomposition of {H^{-1}(\operatorname{div},\Omega)} and construct a new formulation stable on first-order and zero-order Sobolev spaces. Finite element discretisations are then given with respect to the new formulation, and both theoretical analysis and numerical verification are given.

scholarly journals Pedagogia do Projecto

1970 ◽  
Author(s):  
José António Bandeirinha
Keyword(s):  
Order Function

Dos objectivos pedagógicos do ensino da Arquitectura destaca-se um desenvolvimento da capacidade instrumental dos alunos para o exercício do projecto. Mas o projecto aqui enunciado não é uma entidade em si mesma, não existe sem a corporização que lhe é dada, em primeira análise, pelas condições da encomenda — programa, contexto, valor — e, num segundo momento, pela concretização arquitectónica das soluções, cujo principal objectivo é, por estranho que pareça, extinguir esse mesmo projecto, transformando-o em obra. E é, precisamente, à luz desta condição etérea e parasitante do projecto que podem ser colocadas algumas das mais relevantes questões que se lhe põem, na actualidade.   ABSTRACT Given pedagogical scopes of architectural teaching, we can stand out the ability to design practise, in particular for architectural project. But this project we’re writing about should not be considered as an entity in itself. It doesn’t exist without its inherent materialization, given by contextual conditions — order, function, budget — in first place, and, in second place and oddly enough, it doesn’t exist without being extinguished by its transformation in built reality. It’s precisely on behalf of this strange conditions of design practises that we can point some of the most relevant matters on which it lay, today.

Pharmacological Data and Technical Feasibility of Intraperitoneal 5-Fluorouracil Administration

1982 ◽  
Vol 68 (5) ◽  
pp. 437-441 ◽  
Author(s):  
Romano Demicheli ◽  
Antonio Jirillo ◽  
Giorgio Bonciarelli ◽  
Anna Bellini ◽  
Luigi Petrosino ◽  
...  
Keyword(s):  
Malignant Disease ◽  
Technical Feasibility ◽  
Order Function ◽  
Drug Levels ◽  
Tenckhoff Catheter ◽  
Plasma Ratio ◽  
First Order ◽  
Concentration Time ◽  
Pharmacological Data

Via a surgically implanted Tenckhoff catheter, 5-fluorouracil was intraperitoneally administered to patients with malignant disease confined to abdominal space. Treatment was well tolerated without local complications. Peritoneal and plasmatic drug levels were measured, showing that: 1) peritoneal drug levels declined as a first order function; 2) plasmatic levels were very close to those reported for continuous i.v. administration, but peritoneal concentrations were much higher (log 1 to 3); 3) concentration × time product had a peritoneum: plasma ratio ranging from 120 to 1350. The hypothesized role of intraperitoneal 5-fluorouracil administration and the questions still to be answered are summarized.

An optimized constitutive model for reproducing flow stress–strain relationships of anisotropic materials

2018 ◽  
Vol 233 (4) ◽  
pp. 1357-1368 ◽  
Author(s):  
Yanli Lin ◽  
Guannan Chu ◽  
Caiyuan Lin ◽  
Yongda Yan
Keyword(s):  
Flow Stress ◽  
Fourth Order ◽  
Anisotropic Materials ◽  
Second Order ◽  
Least Square ◽  
Experimental Stress ◽  
Function Model ◽  
Stress Strain ◽  
Order Function ◽  
Ordinary Least Square

Due to the strong anisotropic property of the advanced metal materials used in automobile, aviation, and aerospace, experimental flow stress–strain relations including different stress states are necessary to provide the information of anisotropic hardening and plastic flow for constructing a constitutive model. Therefore, reasonably reproducing the experimental stress–strain relations is the most fundamental work to substitute adequate flow stress–strain curves into the constitutive equation at the same time. However, accurate and stable regression results are difficult to obtain through the current regression models such as power exponent, second-order function model, fourth-order function model, and so forth. In this paper, an optimized model named as a least square quadratic regression model (ordinary least square model) was proposed based on the most useful second-order function model. The significant difference is that all experimental points are used to reproduce the experimental stress–strain relations in ordinary least square model in place of only three experimental points adopted in second-order function model, which results in good regression accuracy. Through comparison, it is found that the regression results by power function are poor with regard to some experimental results, and the results reproduced by second-order function model or fourth-order function model are very sensitive to the experimental points selected to do the regression. The sum of squares for error (SSE) increases sharply when the selected points are unreasonable. In addition, for second-order function and fourth-order function models, only limited experimental points are adopted to do the regression, the best regression accuracy cannot be obtained even if the selected points are reasonable. In contrast, SSE of the regression curve by ordinary least square model reduces to less than 50% of the best regressed result by second-order function model, the yielding behavior and variable strain increment ratio of the anisotropic materials can be reflected more accurately. This is very important for accurately describing the plastic flow behaviors of anisotropic materials.

Application of Dynamic Fractional Differentiation to the Study of Oscillating Viscoelastic Medium With Cylindrical Cavity

2002 ◽  
Vol 124 (4) ◽  
pp. 642-645 ◽  
Author(s):  
D. Ingman ◽  
J. Suzdalnitsky
Keyword(s):  
Constitutive Equation ◽  
Fractional Derivative ◽  
Cylindrical Cavity ◽  
Viscoelastic Medium ◽  
Viscoelastic Behavior ◽  
Additional Term ◽  
Variable Order ◽  
Fractional Differentiation ◽  
Order Function ◽  
Damping Effect

Oscillations of a viscoelastic medium with a cylindrical cavity are studied. The viscosity is taken into account in the form of an additional term in the constitutive equation, proportional to a fractional derivative of variable order. In the considered examples the order function corresponds to dependences obtained for real materials. A damping effect is observed in the amplitude behavior. The field which determines the order function demonstrates the viscoelastic behavior of the material under load.

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