1,585 research outputs found
On Basis Constructions in Finite Element Exterior Calculus
We give a systematic and self-contained account of the construction of
geometrically decomposed bases and degrees of freedom in finite element
exterior calculus. In particular, we elaborate upon a previously overlooked
basis for one of the families of finite element spaces, which is of interest
for implementations. Moreover, we give details for the construction of
isomorphisms and duality pairings between finite element spaces. These
structural results show, for example, how to transfer linear dependencies
between canonical spanning sets, or give a new derivation of the degrees of
freedom
Symmetry and Invariant Bases in Finite Element Exterior Calculus
We study symmetries of bases and spanning sets in finite element exterior
calculus using representation theory. The group of affine symmetries of a
simplex is isomorphic to a permutation group and represented on simplicial
finite element spaces by the pullback action. We want to know which
vector-valued finite element spaces have bases that are invariant under
permutation of vertex indices. We determine a natural notion of invariance and
sufficient conditions on the dimension and polynomial degree for the existence
of invariant bases. We conjecture that these conditions are necessary too. We
utilize Djokovic and Malzan's classification of monomial irreducible
representations of the symmetric group and use symmetries of the geometric
decomposition and canonical isomorphisms of the finite element spaces.
Invariant bases are constructed in dimensions two and three for different
spaces of finite element differential forms.Comment: 27 pages. Submitte
Higher-order chain rules for tensor fields, generalized Bell polynomials, and estimates in Orlicz-Sobolev-Slobodeckij and bounded variation spaces
We describe higher-order chain rules for multivariate functions and tensor
fields. We estimate Sobolev-Slobodeckij norms, Musielak-Orlicz norms, and the
total variation seminorms of the higher derivatives of tensor fields after a
change of variables and determine sufficient regularity conditions for the
coordinate change. We also introduce a novel higher-order chain rule for
composition chains of multivariate functions that is described via nested set
partitions and generalized Bell polynomials; it is a natural extension of the
Fa\`a di Bruno formula. Our discussion uses the coordinate-free language of
tensor calculus and includes Fr\'echet-differentiable mappings between Banach
spaces.Comment: Submitte
Smoothed projections over manifolds in finite element exterior calculus
We develop commuting finite element projections over smooth Riemannian
manifolds. This extension of finite element exterior calculus establishes the
stability and convergence of finite element methods for the Hodge-Laplace
equation on manifolds. The commuting projections use localized mollification
operators, building upon a classical construction by de Rham. These projections
are uniformly bounded on Lebesgue spaces of differential forms and map onto
intrinsic finite element spaces defined with respect to an intrinsic smooth
triangulation of the manifold. We analyze the Galerkin approximation error.
Since practical computations use extrinsic finite element methods over
approximate computational manifolds, we also analyze the geometric error
incurred.Comment: Submitted. 31 page
Towards finite element exterior calculus on manifolds: commuting projections, geometric variational crimes, and approximation errors
We survey recent contributions to finite element exterior calculus on
manifolds and surfaces within a comprehensive formalism for the error analysis
of vector-valued partial differential equations on manifolds. Our primary focus
is on uniformly bounded commuting projections on manifolds: these projections
map from Sobolev de Rham complexes onto finite element de Rham complexes,
commute with the differential operators, and satisfy uniform bounds in Lebesgue
norms. They enable the Galerkin theory of Hilbert complexes for a large range
of intrinsic finite element methods on manifolds. However, these intrinsic
finite element methods are generally not computable and thus primarily of
theoretical interest. This leads to our second point: estimating the geometric
variational crime incurred by transitioning to computable approximate problems.
Lastly, our third point addresses how to estimate the approximation error of
the intrinsic finite element method in terms of the mesh size. If the solution
is not continuous, then such an estimate is achieved via modified Cl\'ement or
Scott-Zhang interpolants that facilitate a broken Bramble--Hilbert lemma.Comment: Contribution to ENUMATH Proceedings 2023. 8 page
Corn (\u3ci\u3eZea mays L.\u3c/i\u3e) seeding rate optimization in Iowa, USA
Collecting soil, topography, and yield information has become more feasible and reliable with advancements in precision technologies. Combined with the accessibility of precision technologies and services to farmers, there has been increased interest and ability to make site-specific crop management decisions. The objective of this research was to develop procedures to optimize corn seeding rates and maximize yield using soil and topographic parameters. Experimental treatments included five seeding rates (61 750; 74 100; 86 450; 98 800; and 111 150 seeds ha-1) in a randomized complete block design in three central Iowa fields from 2012 to 2014 (nine site-years). Soil samples were analyzed for available phosphorus (Olsen method), exchangeable potassium (ammonium-acetate method), pH, soil organic matter (SOM), cation exchange capacity (CEC), and texture. Topographic data (in-field elevation, slope, aspect, and curvature) were determined from publically available light detection and ranging data. In four site-years, no interaction occurred between seeding rate and the descriptive variables. Three of the site-years resulted in a negative linear seeding rate response which made it impossible to determine an optimum seeding rate above the lowest seeding rate treatment. The seeding rate optimization process in five site-years resulted in seeding rate by variable interactions; four site-years had a single seeding rate by variable interaction (pH, in-field elevation, or curvature) and one site-year had three seeding rate by variable interactions (pH, CEC, and SOM). Meaningful seeding rate optimizations occurred in only three of nine site-years. There was not a consistent descriptive variable interaction with seeding rate as a result of weather variability. Referenc
Corn (\u3ci\u3eZea mays L.\u3c/i\u3e) seeding rate optimization in Iowa, USA
Collecting soil, topography, and yield information has become more feasible and reliable with advancements in precision technologies. Combined with the accessibility of precision technologies and services to farmers, there has been increased interest and ability to make site-specific crop management decisions. The objective of this research was to develop procedures to optimize corn seeding rates and maximize yield using soil and topographic parameters. Experimental treatments included five seeding rates (61 750; 74 100; 86 450; 98 800; and 111 150 seeds ha-1) in a randomized complete block design in three central Iowa fields from 2012 to 2014 (nine site-years). Soil samples were analyzed for available phosphorus (Olsen method), exchangeable potassium (ammonium-acetate method), pH, soil organic matter (SOM), cation exchange capacity (CEC), and texture. Topographic data (in-field elevation, slope, aspect, and curvature) were determined from publically available light detection and ranging data. In four site-years, no interaction occurred between seeding rate and the descriptive variables. Three of the site-years resulted in a negative linear seeding rate response which made it impossible to determine an optimum seeding rate above the lowest seeding rate treatment. The seeding rate optimization process in five site-years resulted in seeding rate by variable interactions; four site-years had a single seeding rate by variable interaction (pH, in-field elevation, or curvature) and one site-year had three seeding rate by variable interactions (pH, CEC, and SOM). Meaningful seeding rate optimizations occurred in only three of nine site-years. There was not a consistent descriptive variable interaction with seeding rate as a result of weather variability. Referenc
Local Finite Element Approximation of Sobolev Differential Forms
We address fundamental aspects in the approximation theory of vector-valued
finite element methods, using finite element exterior calculus as a unifying
framework. We generalize the Cl\'ement interpolant and the Scott-Zhang
interpolant to finite element differential forms, and we derive a broken
Bramble-Hilbert Lemma. Our interpolants require only minimal smoothness
assumptions and respect partial boundary conditions. This permits us to state
local error estimates in terms of the mesh size. Our theoretical results apply
to curl-conforming and divergence-conforming finite element methods over
simplicial triangulations.Comment: 22 pages. Comments welcom
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Cas9+ conditionally-immortalized macrophages as a tool for bacterial pathogenesis and beyond.
Macrophages play critical roles in immunity, development, tissue repair, and cancer, but studies of their function have been hampered by poorly-differentiated tumor cell lines and genetically-intractable primary cells. Here we report a facile system for genome editing in non-transformed macrophages by differentiating ER-Hoxb8 myeloid progenitors from Cas9-expressing transgenic mice. These conditionally immortalized macrophages (CIMs) retain characteristics of primary macrophages derived from the bone marrow yet allow for easy genetic manipulation and a virtually unlimited supply of cells. We demonstrate the utility of this system for dissection of host genetics during intracellular bacterial infection using two important human pathogens: Listeria monocytogenes and Mycobacterium tuberculosis
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