Title: Variational problems for tree roots and branches
This paper studies two classes of variational problems introduced in Bressan and Sun (On the optimal shape of tree roots and branches. arXiv:1803.01042), related to the optimal shapes of tree roots and branches. Given a measure μ describing the distribution of leaves, a sunlight functional S(μ) computes the total amount of light captured by the leaves. For a measure μ describing the distribution of root hair cells, a harvest functional H(μ) computes the total amount of water and nutrients gathered by the roots. In both cases, we seek a measure μ that maximizes these functionals subject to a ramified transportation cost, for transporting nutrients from the roots to the trunk or from the trunk to the leaves. Compared with Bressan and Sun, here we do not impose any a priori bound on the total mass of the optimal measure μ, and more careful a priori estimates are thus required. In the unconstrained optimization problem for branches, we prove that an optimal measure exists, with bounded support and bounded total mass. In the unconstrained problem for tree roots, we prove that an optimal measure exists, with bounded support but possibly unbounded total mass. The last section of the paper analyzes how the size of the optimal tree depends on the parameters defining the various functionals. more »« less
Bressan, Alberto; Sun, Qing
(, Mathematical Models and Methods in Applied Sciences)
null
(Ed.)
This paper introduces two classes of variational problems, determining optimal shapes for tree roots and branches. Given a measure [Formula: see text], describing the distribution of leaves, we introduce a sunlight functional [Formula: see text] computing the total amount of light captured by the leaves. On the other hand, given a measure [Formula: see text] describing the distribution of root hair cells, we consider a harvest functional [Formula: see text] computing the total amount of water and nutrients gathered by the roots. In both cases, we seek to maximize these functionals subject to a ramified transportation cost, for transporting nutrients from the roots to the trunk and from the trunk to the leaves. The main results establish various properties of these functionals, and the existence of optimal distributions. In particular, we prove the upper semicontinuity of [Formula: see text] and [Formula: see text], together with a priori estimates on the support of optimal distributions.
M.C. Lastinger, A. Rice
(, IEEE Aerospace Conference)
We discuss a novel variable topology continuum robot designed and constructed under NASA-funded research. The robot features a seven degree of freedom continuous backbone “trunk”, with two pairs of “branches”: two “tendril” effectors and two support “roots”. Each of the pairs of branches can be fully retracted inside the trunk, allowing it to penetrate congested environments as a single slender unit, and subsequently deploy the branches to perform a variety of tasks. The “roots” provide physical support, while the two effectors and trunk tip enable independent but coordinated functionality: sensing (vision) in one tendril, and manipulation at two scales, via the second tendril and the trunk tip. The specifics of the new design are described and discussed in detail. We illustrate the operation and potential applications of the new design via a series of demonstrations, particularly cleaning of dust from solar panels.
Crous, PW; Costa, MM; Kandemir, H; Vermaas, M; Vu, D; Zhao, L; Arumugam, E; Flakus, A; Jurjević, Ž; Kaliyaperumal, M; et al
(, Persoonia - Molecular Phylogeny and Evolution of Fungi)
Crous, Pedro
(Ed.)
Novel species of fungi described in this study include those from various countries as follows: Argentina, Neocamarosporium halophilum in leaf spots of Atriplex undulata. Australia, Aschersonia merianiae on scale insect (Coccoidea), Curvularia huamulaniae isolated from air, Hevansia mainiae on dead spider, Ophiocordyceps poecilometigena on Poecilometis sp. Bolivia, Lecanora menthoides on sandstone, in open semi-desert montane areas, Sticta monlueckiorum corticolous in a forest, Trichonectria epimegalosporae on apothecia of corticolous Megalospora sulphurata var. sulphurata, Trichonectria puncteliae on the thallus of Punctelia borreri. Brazil, Catenomargarita pseudocercosporicola (incl. Catenomargarita gen. nov.) hyperparasitic on Pseudocercospora fijiensis on leaves of Musa acuminata, Tulasnella restingae on protocorms and roots of Epidendrum fulgens. Bulgaria, Anthracoidea umbrosae on Carex spp. Croatia, Hymenoscyphus radicis from surface-sterilised, asymptomatic roots of Microthlaspi erraticum, Orbilia multiserpentina on wood of decorticated branches of Quercus pubescens. France, Calosporella punctatispora on dead corticated twigs of Acer opalus. French West Indies (Martinique), Eutypella lechatii on dead corticated palm stem. Germany, Arrhenia alcalinophila on loamy soil. Iceland, Cistella blauvikensis on dead grass (Poaceae). India, Fulvifomes maritimus on living Peltophorum pterocarpum, Fulvifomes natarajanii on dead wood of Prosopis juliflora, Fulvifomes subazonatus on trunk of Azadirachta indica, Macrolepiota bharadwajii on moist soil near the forest, Narcissea delicata on decaying elephant dung, Paramyrothecium indicum on living leaves of Hibiscus hispidissimus, Trichoglossum syamviswanathii on moist soil near the base of a bamboo plantation. Iran, Vacuiphoma astragalicola from stem canker of Astragalus sarcocolla. Malaysia, Neoeriomycopsis fissistigmae (incl. Neoeriomycopsidaceae fam. nov.) on leaf spots on flower Fissistigma sp. Namibia, Exophiala lichenicola lichenicolous on Acarospora cf. luederitzensis. Netherlands, Entoloma occultatum on soil, Extremus caricis on dead leaves of Carex sp., Inocybe pseudomytiliodora on loamy soil. Norway, Inocybe guldeniae on calcareous soil, Inocybe rupestroides on gravelly soil. Pakistan, Hymenagaricus brunneodiscus on soil. Philippines, Ophiocordyceps philippinensis parasitic on Asilus sp. Poland, Hawksworthiomyces ciconiae isolated from Ciconia ciconia nest, Plectosphaerella vigrensis from leaf spots on Impatiens noli-tangere, Xenoramularia epitaxicola from sooty mould community on Taxus baccata. Portugal, Inocybe dagamae on clay soil. Saudi Arabia, Diaporthe jazanensis on branches of Coffea arabica. South Africa, Alternaria moraeae on dead leaves of Moraea sp., Bonitomyces buffelskloofinus (incl. Bonitomyces gen. nov.) on dead twigs of unknown tree, Constrictochalara koukolii on living leaves of Itea rhamnoides colonised by a Meliola sp., Cylindromonium lichenophilum on Parmelina tiliacea, Gamszarella buffelskloofina (incl. Gamszarella gen. nov.) on dead insect, Isthmosporiella africana (incl. Isthmosporiella gen. nov.) on dead twigs of unknown tree, Nothoeucasphaeria buffelskloofina (incl. Nothoeucasphaeria gen. nov.), on dead twigs of unknown tree, Nothomicrothyrium beaucarneae (incl. Nothomicrothyrium gen. nov.) on dead leaves of Beaucarnea stricta, Paramycosphaerella proteae on living leaves of Protea caffra, Querciphoma foliicola on leaf litter, Rachicladosporium conostomii on dead twigs of Conostomium natalense var. glabrum, Rhamphoriopsis synnematosa on dead twig of unknown tree, Waltergamsia mpumalanga on dead leaves of unknown tree. Spain, Amanita fulvogrisea on limestone soil, in mixed forest, Amanita herculis in open Quercus forest, Vuilleminia beltraniae on Cistus symphytifolius. Sweden, Pachyella pulchella on decaying wood on sand-silt riverbank. Thailand, Deniquelata cassiae on dead stem of Cassia fistula, Stomiopeltis thailandica on dead twigs of Magnolia champaca. Ukraine, Circinaria podoliana on natural limestone outcrops, Neonematogonum carpinicola (incl. Neonematogonum gen. nov.) on dead branches of Carpinus betulus. USA, Exophiala wilsonii water from cooling tower, Hygrophorus aesculeticola on soil in mixed forest, and Neocelosporium aereum from air in a house attic. Morphological and culture characteristics are supported by DNA barcodes.
Abstract Wind-induced stress is the primary mechanical cause of tree failures. Among different factors, the branching mechanism plays a central role in the stress distribution and stability of trees in windstorms. A recent study showed that Leonardo da Vinci’s original observation, stating that the total cross section of branches conserved across branching nodes is the optimal configuration for resisting wind-induced damage in rigid trees, is correct. However, the breaking risk and the optimal branching pattern of trees are also a function of their reconfiguration capabilities and the processes they employ to mitigate high wind-induced stress hotspots. In this study, using a numerical model of rigid and flexible branched trees, we explore the role of flexibility and branching patterns of trees in their reconfiguration and stress mitigation capabilities. We identify the robust optimal branching mechanism for an extensive range of tree flexibility. Our results show that the probability of a tree breaking at each branching level from the stem to terminal foliage strongly depends on the cross section changes in the branching nodes, the overall tree geometry, and the level of tree flexibility. Three response categories have been identified: the stress concentration in the main trunk, the uniform stress level through the tree’s height, and substantial stress localization in the terminal branches. The reconfigurability of the tree determines the dominant response mode. The results suggest a very similar optimal branching law for both flexible and rigid trees wherein uniform stress distribution occurs throughout the tree’s height. An exception is the very flexible branched plants in which the optimal branching pattern deviates from this prediction and is strongly affected by the reconfigurability of the tree.
Snelson, Stanley; Teixeira, Eduardo V
(, SIAM Journal on Mathematical Analysis)
We prove the existence of an open set minimizing the first Dirichlet eigenvalue of an elliptic operator with bounded, measurable coefficients, over all open sets of a given measure. Our proof is based on a free boundary approach: we characterize the eigenfunction on the optimal set as the minimizer of a penalized functional, and derive openness of the optimal set as a consequence of a Hölder estimate for the eigenfunction. We also prove that the optimal eigenfunction grows at most linearly from the free boundary, i.e., it is Lipschitz continuous at free boundary points.
Bressan, Alberto, Palladino, Michele, and Sun, Qing. Variational problems for tree roots and branches. Retrieved from https://par.nsf.gov/biblio/10162554. Calculus of variations and partial differential equations 59.7 Web. doi:10.1007/s00526-019-1666-1.
Bressan, Alberto, Palladino, Michele, & Sun, Qing. Variational problems for tree roots and branches. Calculus of variations and partial differential equations, 59 (7). Retrieved from https://par.nsf.gov/biblio/10162554. https://doi.org/10.1007/s00526-019-1666-1
@article{osti_10162554,
place = {Country unknown/Code not available},
title = {Variational problems for tree roots and branches},
url = {https://par.nsf.gov/biblio/10162554},
DOI = {10.1007/s00526-019-1666-1},
abstractNote = {This paper studies two classes of variational problems introduced in Bressan and Sun (On the optimal shape of tree roots and branches. arXiv:1803.01042), related to the optimal shapes of tree roots and branches. Given a measure μ describing the distribution of leaves, a sunlight functional S(μ) computes the total amount of light captured by the leaves. For a measure μ describing the distribution of root hair cells, a harvest functional H(μ) computes the total amount of water and nutrients gathered by the roots. In both cases, we seek a measure μ that maximizes these functionals subject to a ramified transportation cost, for transporting nutrients from the roots to the trunk or from the trunk to the leaves. Compared with Bressan and Sun, here we do not impose any a priori bound on the total mass of the optimal measure μ, and more careful a priori estimates are thus required. In the unconstrained optimization problem for branches, we prove that an optimal measure exists, with bounded support and bounded total mass. In the unconstrained problem for tree roots, we prove that an optimal measure exists, with bounded support but possibly unbounded total mass. The last section of the paper analyzes how the size of the optimal tree depends on the parameters defining the various functionals.},
journal = {Calculus of variations and partial differential equations},
volume = {59},
number = {7},
author = {Bressan, Alberto and Palladino, Michele and Sun, Qing},
}
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