A staggered approach for the coupling of Cahn–Hilliard type diffusion and finite strain elasticity

We develop an algorithm and computational implementation for simulation of problems that combine Cahn–Hilliard type diffusion with finite strain elasticity. We have in mind applications such as the electro-chemo- mechanics of lithium ion (Li-ion) batteries. We concentrate on basic computational aspects. A staggered algorithm is pro- posed for the coupled multi-field model. For the diffusion problem, the fourth order differential equation is replaced by a system of second order equations to deal with the issue of the regularity required for the approximation spaces. Low order finite elements are used for discretization in space of the involved fields (displacement, concentration, nonlocal concentration). Three (both 2D and 3D) extensively worked numerical examples show the capabilities of our approach for the representation of (i) phase separation, (ii) the effect of concentration in deformation and stress, (iii) the effect of Electronic supplementary material The online version of this article (doi:10.1007/s00466-015-1235-1) contains supplementary material, which is available to authorized users. B P. Areias pmaa@uevora.pt 1 Department of Physics, University of Évora, Colégio Luís António Verney, Rua Romão Ramalho, 59, 7002-554 Évora, Portugal 2 ICIST, Lisbon, Portugal 3 School of Engineering, Universidad de Cuenca, Av. 12 de Abril s/n. 01-01-168, Cuenca, Ecuador 4 Institute of Structural Mechanics, Bauhaus-University Weimar, Marienstraße 15, 99423 Weimar, Germany strain in concentration, and (iv) lithiation. We analyze con- vergence with respect to spatial and time discretization and found that very good results are achievable using both a stag- gered scheme and approximated strain interpolation.

Effects of substrate type on plant growth and nitrogen and nitrate concentration in spinach

Abstract The effects of three commercial substrates (a mixture of forest residues, composted grape husks, and white peat, black peat and coir) on plant growth and nitrogen (N) and nitrate (NO3) concentration and content were evaluated in spinach (Spinacia oleracea L. cv. Tapir). Spinach seedlings were transplanted at 45 days after emergence into Styrofoam boxes filled with the substrates and were grown during winter and early spring in an unheated greenhouse with no supplemental lighting. Each planting box was irrigated daily by drip and fertilized with a complete nutrient solution. The NO3 content of the drainage water was lower in coir than in the other substrates. However, shoot NO3 concentration was not affected by substrate type, while yield and total shoot N and NO3 content were greater when plants were grown in peat than in the mixed substrate or the coir. Leaf chlorophyll meter readings provided a good indication of the amount of N in the plants and increased linearly with total shoot N. Keywords Spinacia oleracea; chlorophyll meter; coir; peat; soilless culture systems

Effects of substrate type on plant growth and nitrogen and nitrate concentration in spinach.

The effects of three commercial substrates (a mixture of forest residues, composted grape husks, and white peat, black peat and coir) on plant growth and nitrogen (N) and nitrate (NO3) concentration and content were evaluated in spinach (Spinacia oleracea L. cv. Tapir). Spinach seedlings were transplanted at 45 days after emergence into Styrofoam boxes filled with the substrates and were grown during winter and early spring in an unheated greenhouse with no supplemental lighting. Each planting box was irrigated daily by drip and fertilized with a complete nutrient solution. The NO3 content of the drainage water was lower in coir than in the other substrates. However, shoot NO3 concentration was not affected by substrate type, while yield and total shoot N and NO3 content were greater when plants were grown in peat than in the mixed substrate or the coir. Leaf chlorophyll meter readings provided a good indication of the amount of N in the plants and increased linearly with total shoot N.

Transition matrices characterizing a certain totally discontinuous map of the interval

We study a totally discontinuous interval map defined in [0,1] which is associated to a deformation of the shift map on two symbols 0−1. We define a sequence of transition matrices which characterizes the effect of the interval map on a family of partitions of the interval [0,1]. Recursive algorithms that build the sequence of matrices and their left and right eigenvectors are deduced. Moreover, we compute the Artin zeta function for the interval map.

Phase-field analysis of finite-strain plates and shells including element subdivision

With the theme of fracture of finite-strain plates and shells based on a phase-field model of crack regularization, we introduce a new staggered algorithm for elastic and elasto-plastic materials. To account for correct fracture behavior in bending, two independent phase-fields are used, corresponding to the lower and upper faces of the shell. This is shown to provide a realistic behavior in bending-dominated problems, here illustrated in classical beam and plate problems. Finite strain behavior for both elastic and elasto-plastic constitutive laws is made compatible with the phase-field model by use of a consistent updated-Lagrangian algorithm. To guarantee sufficient resolution in the definition of the crack paths, a local remeshing algorithm based on the phase- field values at the lower and upper shell faces is introduced. In this local remeshing algorithm, two stages are used: edge-based element subdivision and node repositioning. Five representative numerical examples are shown, consisting of a bi-clamped beam, two versions of a square plate, the Keesecker pressurized cylinder problem, the Hexcan problem and the Muscat-Fenech and Atkins plate. All problems were successfully solved and the proposed solution was found to be robust and efficient.