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Piola Transformation
The Piola transformation maps vectors between Eulerian and Lagrangian coordinates in continuum mechanics. It is named after Gabrio Piola. Definition Let F: \mathbb^d \rightarrow \mathbb^d with F( \hat) = B \hat +b, ~ B \in \mathbb^, ~ b \in \mathbb^ an affine transformation. Let K=F(\hat) with \hat a domain with Lipschitz boundary. The mapping p: L^2( \hat )^d \rightarrow L^2(K)^d, \quad \hat \mapsto p(\hat)(x) := \frac \cdot B \hat (\hat) is called Piola transformation. The usual definition takes the absolute value of the determinant, although some authors make it just the determinant. Note: for a more general definition in the context of tensors and elasticity, as well as a proof of the property that the Piola transform conserves the flux of tensor fields across boundaries, see Ciarlet's book. See also * Piola–Kirchhoff stress tensor *Raviart–Thomas basis functions In applied mathematics, Raviart–Thomas basis functions are vector basis functions used in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrangian And Eulerian Specification Of The Flow Field
__NOTOC__ In classical field theories, the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time. Plotting the position of an individual parcel through time gives the pathline of the parcel. This can be visualized as sitting in a boat and drifting down a river. The Eulerian specification of the flow field is a way of looking at fluid motion that focuses on specific locations in the space through which the fluid flows as time passes. This can be visualized by sitting on the bank of a river and watching the water pass the fixed location. The Lagrangian and Eulerian specifications of the flow field are sometimes loosely denoted as the Lagrangian and Eulerian frame of reference. However, in general both the Lagrangian and Eulerian specification of the flow field can be applied in any observer's frame of reference, and in any coordinate system used within the chosen fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Continuum Mechanics
Continuum mechanics is a branch of mechanics that deals with the mechanical behavior of materials modeled as a continuous mass rather than as discrete particles. The French mathematician Augustin-Louis Cauchy was the first to formulate such models in the 19th century. Explanation A continuum model assumes that the substance of the object fills the space it occupies. Modeling objects in this way ignores the fact that matter is made of atoms, and so is not continuous; however, on length scales much greater than that of inter-atomic distances, such models are highly accurate. These models can be used to derive differential equations that describe the behavior of such objects using physical laws, such as mass conservation, momentum conservation, and energy conservation, and some information about the material is provided by constitutive relationships. Continuum mechanics deals with the physical properties of solids and fluids which are independent of any particular coordinate sy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gabrio Piola
Gabrio Piola (15 July 1794 – 9 November 1850) was an Italian mathematician and physicist, Danilo Capecchi and Giuseppe C. Ruta"Piola's contribution to continuum mechanics" ''Archive for History of Exact Sciences'', Vol. 61, No. 4 (July 2007), pp. 303-342 member of the Lombardo Institute of Science, Letters and Arts. He studied in particular the mechanics of the continuous, linking his name to the tensors called Piola–Kirchhoff. Biography Count Gabrio Piola Daverio was born in Milan in a rich and aristocratic family. Initially he studied at home and then at the local high school. Given his exceptional ability in mathematics and physics, he started to study mathematics at the University of Pavia, as a student of Vincenzo Brunacci, obtaining his doctorate on 24 June 1816. He didn’t follow an academic career even though he was offered the chair of Applied Mathematics in Rome; he preferred dedicating himself to private teaching. One of his students was Francesco Brioschi w ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Philippe G
Philippe is a masculine sometimes feminin given name, cognate to Philip. It may refer to: * Philippe of Belgium (born 1960), King of the Belgians (2013–present) * Philippe (footballer) (born 2000), Brazilian footballer * Prince Philippe, Count of Flanders, father to Albert I of Belgium * Philippe d'Orléans (other), multiple people * Philippe A. Autexier (1954–1998), French music historian * Philippe Blain, French volleyball player and coach * Philippe Najib Boulos (1902–1979), Lebanese lawyer and politician * Philippe Coutinho, Brazilian footballer * Philippe Daverio (1949–2020), Italian art historian * Philippe Dubuisson-Lebon, Canadian football player * Philippe Ginestet (born 1954), French billionaire businessman, founder of GiFi * Philippe Gilbert, Belgian bicycle racer * Philippe Petit, French performer and tightrope artist * Philippe Petitcolin (born 1952/53), French businessman, CEO of Safran * Philippe Russo, French singer * Philippe Sella, French rugby pla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Piola–Kirchhoff Stress Tensor
In continuum mechanics, stress is a physical quantity. It is a quantity that describes the magnitude of forces that cause deformation. Stress is defined as ''force per unit area''. When an object is pulled apart by a force it will cause elongation which is also known as deformation, like the stretching of an elastic band, it is called tensile stress. But, when the forces result in the compression of an object, it is called compressive stress. It results when forces like tension or compression act on a body. The greater this force and the smaller the cross-sectional area of the body on which it acts, the greater the stress. Therefore, stress is measured in newton per square meter (N/m2) or pascal (Pa). Stress expresses the internal forces that neighbouring particles of a continuous material exert on each other, while strain is the measure of the deformation of the material. For example, when a solid vertical bar is supporting an overhead weight, each particle in the bar pushe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Raviart–Thomas Basis Functions
In applied mathematics, Raviart–Thomas basis functions are vector basis functions used in finite element and boundary element methods. They are regularly used as basis functions when working in electromagnetics. They are sometimes called Rao-Wilton-Glisson basis functions. The space \mathrm_q spanned by the Raviart–Thomas basis functions of order q is the smallest polynomial space such that the divergence In vector calculus, divergence is a vector operator that operates on a vector field, producing a scalar field giving the quantity of the vector field's source at each point. More technically, the divergence represents the volume density of the ... maps \mathrm_q onto \mathrm_q, the space of piecewise polynomials of order q. Order 0 Raviart-Thomas Basis Functions in 2D In two-dimensional space, the lowest order Raviart Thomas space, \mathrm_0, has degrees of freedom on the edges of the elements of the finite element mesh. The nth edge has an associated basis function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
