We advocate a simple geometric model for elasticity: distance between the differential of a deformation and the rotation group. It comes with rigorous differential geometric underpinnings, both smooth and discrete, and is computationally almost as simple and efficient as linear elasticity. Owing to its geometric non-linearity, though, it does not suffer from the usual linearization artifacts. A material model with standard elastic moduli (Lamé parameters) falls out naturally, and a minimizer for static problems is easily augmented to construct a fully variational 2nd order time integrator. It has excellent conservation properties even for very coarse simulations, making it very robust.Our analysis was motivated by a number of heuristic, physics-like algorithms from geometry processing (editing, morphing, parameterization, and simulation). Starting with a continuous energy formulation and taking the underlying geometry into account, we simplify and accelerate these algorithms while avoiding common pitfalls. Through the connection with the Biot strain of mechanics, the intuition of previous work that these ideas are "like" elasticity is shown to be spot on.Keywords: Digital Geometry Processing, Discrete Differential Geometry, elasticity, geometric modeling, shape space interpolation, morphing, parameterization. The Elastic EnergyWe jump right into the meat of things and defer discussion of relations to previous algorithms until we complete our setup.Given a smooth map f : M →M , f (p) = q describing the deformation of a reference configuration M ⊂ n into a deformed configurationM ⊂ n , we study the energyAt a point p ∈ M , the integrand measures the distance between the deformation differential, df , and the nearest rotation, thus characterizing how far f is from an isometry. (Our df corresponds to what is often called the deformation gradient F in mechanics.) In 2D this is the setting of planar morphing and parameterization while the 3D case principally covers solid mechanics.A minimizer of E( f ) subject to boundary conditions is characterized by vanishing variationsLet g be an arbitrary admissible variation. Then,where 〈A, B〉 = tr(A T B) denotes the standard inner product between linear maps and R ∈ SO(n) the minimizer of the squared distance (dropping explicit mention of its dependence on df ). We have used df − R ⊥ δ g R, which follows from R being critical with respect to the squared distance; thus, no derivatives of R appear in the gradient of E.Due to the dependence of R on df , the Euler-Lagrange equation is a non-linear Poisson problemThe energy Hessian follows from taking a further variation hThe first term is the standard Laplace-Beltrami operator which does not depend on f , while the second term varies with f through the dependence of R on df . Here variations of R do enter since they are in general not orthogonal to dg.

We advocate a simple geometric model for elasticity: distance between the differential of a deformation and the rotation group. It comes with rigorous differential geometric underpinnings, both smooth and discrete, and is computationally almost as simple and efficient as linear elasticity. Owing to its geometric non-linearity, though, it does not suffer from the usual linearization artifacts. A material model with standard elastic moduli (Lamé parameters) falls out naturally, and a minimizer for static problems is easily augmented to construct a fully variational 2nd order time integrator. It has excellent conservation properties even for very coarse simulations, making it very robust.Our analysis was motivated by a number of heuristic, physics-like algorithms from geometry processing (editing, morphing, parameterization, and simulation). Starting with a continuous energy formulation and taking the underlying geometry into account, we simplify and accelerate these algorithms while avoiding common pitfalls. Through the connection with the Biot strain of mechanics, the intuition of previous work that these ideas are "like" elasticity is shown to be spot on.Keywords: Digital Geometry Processing, Discrete Differential Geometry, elasticity, geometric modeling, shape space interpolation, morphing, parameterization. The Elastic EnergyWe jump right into the meat of things and defer discussion of relations to previous algorithms until we complete our setup.Given a smooth map f : M →M , f (p) = q describing the deformation of a reference configuration M ⊂ n into a deformed configurationM ⊂ n , we study the energyAt a point p ∈ M , the integrand measures the distance between the deformation differential, df , and the nearest rotation, thus characterizing how far f is from an isometry. (Our df corresponds to what is often called the deformation gradient F in mechanics.) In 2D this is the setting of planar morphing and parameterization while the 3D case principally covers solid mechanics.A minimizer of E( f ) subject to boundary conditions is characterized by vanishing variationsLet g be an arbitrary admissible variation. Then,where 〈A, B〉 = tr(A T B) denotes the standard inner product between linear maps and R ∈ SO(n) the minimizer of the squared distance (dropping explicit mention of its dependence on df ). We have used df − R ⊥ δ g R, which follows from R being critical with respect to the squared distance; thus, no derivatives of R appear in the gradient of E.Due to the dependence of R on df , the Euler-Lagrange equation is a non-linear Poisson problemThe energy Hessian follows from taking a further variation hThe first term is the standard Laplace-Beltrami operator which does not depend on f , while the second term varies with f through the dependence of R on df . Here variations of R do enter since they are in general not orthogonal to dg.

We advocate a simple geometric model for elasticity: distance between the differential of a deformation and the rotation group. It comes with rigorous differential geometric underpinnings, both smooth and discrete, and is computationally almost as simple and efficient as linear elasticity. Owing to its geometric non-linearity, though, it does not suffer from the usual linearization artifacts. A material model with standard elastic moduli (Lamé parameters) falls out naturally, and a minimizer for static problems is easily augmented to construct a fully variational 2nd order time integrator. It has excellent conservation properties even for very coarse simulations, making it very robust.Our analysis was motivated by a number of heuristic, physics-like algorithms from geometry processing (editing, morphing, parameterization, and simulation). Starting with a continuous energy formulation and taking the underlying geometry into account, we simplify and accelerate these algorithms while avoiding common pitfalls. Through the connection with the Biot strain of mechanics, the intuition of previous work that these ideas are "like" elasticity is shown to be spot on.Keywords: Digital Geometry Processing, Discrete Differential Geometry, elasticity, geometric modeling, shape space interpolation, morphing, parameterization. The Elastic EnergyWe jump right into the meat of things and defer discussion of relations to previous algorithms until we complete our setup.Given a smooth map f : M →M , f (p) = q describing the deformation of a reference configuration M ⊂ n into a deformed configurationM ⊂ n , we study the energyAt a point p ∈ M , the integrand measures the distance between the deformation differential, df , and the nearest rotation, thus characterizing how far f is from an isometry. (Our df corresponds to what is often called the deformation gradient F in mechanics.) In 2D this is the setting of planar morphing and parameterization while the 3D case principally covers solid mechanics.A minimizer of E( f ) subject to boundary conditions is characterized by vanishing variationsLet g be an arbitrary admissible variation. Then,where 〈A, B〉 = tr(A T B) denotes the standard inner product between linear maps and R ∈ SO(n) the minimizer of the squared distance (dropping explicit mention of its dependence on df ). We have used df − R ⊥ δ g R, which follows from R being critical with respect to the squared distance; thus, no derivatives of R appear in the gradient of E.Due to the dependence of R on df , the Euler-Lagrange equation is a non-linear Poisson problemThe energy Hessian follows from taking a further variation hThe first term is the standard Laplace-Beltrami operator which does not depend on f , while the second term varies with f through the dependence of R on df . Here variations of R do enter since they are in general not orthogonal to dg.

Grid computing has recently become an important paradigm for managing computationally demanding applications, composed of a collection of services. The dynamic discovery of services, and the selection of a particular service instance providing the best value out of the discovered alternatives, poses a complex multi-attribute n:m allocation decision problem, which is often solved using a centralized resource broker. To manage complexity, this article proposes a two-layer architecture for service discovery in such Application Layer Networks (ALN). The first layer consists of a service market in which complex services are translated to a set of basic services, which are distinguished by price and availability. The second layer provides an allocation of services to appropriate resources in order to enact the specified services. This framework comprises the foundations for a later comparison of centralized and decentralized market mechanisms for allocation of services and resources in ALNs and Grids.

In this paper we describe an application deployment using a Catallactic Grid-enabled middleware, which is based on the Catallaxy "free market" self-organisation approach described by von Hayek [7], who understood the market as a decentralised coordination mechanism opposite to a centralised command economy. The implementation makes use of Globus Toolkit, JXTA and WSRF. The paper envisages the resource virtualization in the WSRF context as the main driver for a proper connection middleware-base platform (on the broad scenario of grid applications).