Gradients in urban material composition: A new concept to map cities with spaceborne imaging spectroscopy data

To understand processes in urban environments, such as urban energy fluxes or surface temperature patterns, it is important to map urban surface materials. Airborne imaging spectroscopy data have been successfully used to identify urban surface materials mainly based on unmixing algorithms. Upcoming spaceborne Imaging Spectrometers (IS), such as the Environmental Mapping and Analysis Program (EnMAP), will reduce the time and cost-critical limitations of airborne systems for Earth Observation (EO). However, the spatial resolution of all operated and planned IS in space will not be higher than 20 to 30 m and, thus, the detection of pure Endmember (EM) candidates in urban areas, a requirement for spectral unmixing, is very limited. Gradient analysis could be an alternative method for retrieving urban surface material compositions in pixels from spaceborne IS. The gradient concept is well known in ecology to identify plant species assemblages formed by similar environmental conditions but has never been tested for urban materials. However, urban areas also contain neighbourhoods with similar physical, compositional and structural characteristics. Based on this assumption, this study investigated (1) whether cover fractions of surface materials change gradually in urban areas and (2) whether these gradients can be adequately mapped and interpreted using imaging spectroscopy data (e.g. EnMAP) with 30 m spatial resolution. Similarities of material compositions were analysed on the basis of 153 systematically distributed samples on a detailed surface material map using Detrended Correspondence Analysis (DCA). Determined gradient scores for the first two gradients were regressed against the corresponding mean reflectance of simulated EnMAP spectra using Partial Least Square regression models. Results show strong correlations with R2 = 0.85 and R2 = 0.71 and an RMSE of 0.24 and 0.21 for the first and second axis, respectively. The subsequent mapping of the first gradient reveals patterns that correspond to the transition from predominantly vegetation classes to the dominance of artificial materials. Patterns resulting from the second gradient are associated with surface material compositions that are related to finer structural differences in urban structures. The composite gradient map shows patterns of common surface material compositions that can be related to urban land use classes such as Urban Structure Types (UST). By linking the knowledge of typical material compositions with urban structures, gradient analysis seems to be a powerful tool to map characteristic material compositions in 30 m imaging spectroscopy data of urban areas

Analytical continuum mechanics \`a la Hamilton-Piola: least action principle for second gradient continua and capillary fluids

In this paper a stationary action principle is proven to hold for capillary fluids, i.e. fluids for which the deformation energy has the form suggested, starting from molecular arguments, for instance by Cahn and Hilliard. Remark that these fluids are sometimes also called Korteweg-de Vries or Cahn-Allen. In general continua whose deformation energy depend on the second gradient of placement are called second gradient (or Piola-Toupin or Mindlin or Green-Rivlin or Germain or second gradient) continua. In the present paper, a material description for second gradient continua is formulated. A Lagrangian action is introduced in both material and spatial description and the corresponding Euler-Lagrange bulk and boundary conditions are found. These conditions are formulated in terms of an objective deformation energy volume density in two cases: when this energy is assumed to depend on either C and grad C or on C^-1 and grad C^-1 ; where C is the Cauchy-Green deformation tensor. When particularized to energies which characterize fluid materials, the capillary fluid evolution conditions (see e.g. Casal or Seppecher for an alternative deduction based on thermodynamic arguments) are recovered. A version of Bernoulli law valid for capillary fluids is found and, in the Appendix B, useful kinematic formulas for the present variational formulation are proposed. Historical comments about Gabrio Piola's contribution to continuum analytical mechanics are also presented. In this context the reader is also referred to Capecchi and Ruta.Comment: 52 page

Finite strain viscoplasticity with nonlinear kinematic hardening: phenomenological modeling and time integration

This article deals with a viscoplastic material model of overstress type. The model is based on a multiplicative decomposition of the deformation gradient into elastic and inelastic part. An additional multiplicative decomposition of inelastic part is used to describe a nonlinear kinematic hardening of Armstrong-Frederick type. Two implicit time-stepping methods are adopted for numerical integration of evolution equations, such that the plastic incompressibility constraint is exactly satisfied. The first method is based on the tensor exponential. The second method is a modified Euler-Backward method. Special numerical tests show that both approaches yield similar results even for finite inelastic increments. The basic features of the material response, predicted by the material model, are illustrated with a series of numerical simulations.Comment: 29 pages, 7 figure

Gedanken experiments for the determination of two-dimensional linear second gradient elasticity coefficients

In the present paper, a two-dimensional solid consisting of a linear elastic isotropic material, for which the deformation energy depends on the second gradient of the displacement, is considered. The strain energy is demonstrated to depend on 6 constitutive parameters: the 2 Lam´e constants (λ and μ) and 4 more parameters (instead of 5 as it is in the 3D-case). Analytical solutions for classical problems such as heavy sheet, bending and flexure are provided. The idea is very simple: The solutions of the corresponding problem of first gradient classical case are imposed, and the corresponding forces, double forces and wedge forces are found. On the basis of such solutions, a method is outlined, which is able to identify the six constitutive parameters. Ideal (or Gedanken) experiments are designed in order to write equations having as unknowns the six constants and as known terms the values of suitable experimental measurements

A Subgradient Method for Free Material Design

A small improvement in the structure of the material could save the manufactory a lot of money. The free material design can be formulated as an optimization problem. However, due to its large scale, second-order methods cannot solve the free material design problem in reasonable size. We formulate the free material optimization (FMO) problem into a saddle-point form in which the inverse of the stiffness matrix A(E) in the constraint is eliminated. The size of A(E) is generally large, denoted as N by N. This is the first formulation of FMO without A(E). We apply the primal-dual subgradient method [17] to solve the restricted saddle-point formula. This is the first gradient-type method for FMO. Each iteration of our algorithm takes a total of $O(N^2)$ foating-point operations and an auxiliary vector storage of size O(N), compared with formulations having the inverse of A(E) which requires $O(N^3)$ arithmetic operations and an auxiliary vector storage of size $O(N^2)$. To solve the problem, we developed a closed-form solution to a semidefinite least squares problem and an efficient parameter update scheme for the gradient method, which are included in the appendix. We also approximate a solution to the bounded Lagrangian dual problem. The problem is decomposed into small problems each only having an unknown of k by k (k = 3 or 6) matrix, and can be solved in parallel. The iteration bound of our algorithm is optimal for general subgradient scheme. Finally we present promising numerical results.Comment: SIAM Journal on Optimization (accepted

The 1/r singularity in weakly nonlinear fracture mechanics

Material failure by crack propagation essentially involves a concentration of large displacement-gradients near a crack's tip, even at scales where no irreversible deformation and energy dissipation occurs. This physical situation provides the motivation for a systematic gradient expansion of general nonlinear elastic constitutive laws that goes beyond the first order displacement-gradient expansion that is the basis for linear elastic fracture mechanics (LEFM). A weakly nonlinear fracture mechanics theory was recently developed by considering displacement-gradients up to second order. The theory predicts that, at scales within a dynamic lengthscale $\ell$ from a crack's tip, significant $\log{r}$ displacements and $1/r$ displacement-gradient contributions arise. Whereas in LEFM the $1/r$ singularity generates an unbalanced force and must be discarded, we show that this singularity not only exists but is {\em necessary} in the weakly nonlinear theory. The theory generates no spurious forces and is consistent with the notion of the autonomy of the near-tip nonlinear region. The J-integral in the weakly nonlinear theory is also shown to be path-independent, taking the same value as the linear elastic J-integral. Thus, the weakly nonlinear theory retains the key tenets of fracture mechanics, while providing excellent quantitative agreement with measurements near the tip of single propagating cracks. As $\ell$ is consistent with lengthscales that appear in crack tip instabilities, we suggest that this theory may serve as a promising starting point for resolving open questions in fracture dynamics.Comment: 12 pages, 2 figure