216 research outputs found
Summation-By-Parts Operators and High-Order Quadrature
Summation-by-parts (SBP) operators are finite-difference operators that mimic
integration by parts. This property can be useful in constructing energy-stable
discretizations of partial differential vequations. SBP operators are defined
by a weight matrix and a difference operator, with the latter designed to
approximate to a specified order of accuracy. The accuracy of the weight
matrix as a quadrature rule is not explicitly part of the SBP definition. We
show that SBP weight matrices are related to trapezoid rules with end
corrections whose accuracy matches the corresponding difference operator at
internal nodes. The accuracy of SBP quadrature extends to curvilinear domains
provided the Jacobian is approximated with the same SBP operator used for the
quadrature. This quadrature has significant implications for SBP-based
discretizations; for example, the discrete norm accurately approximates the
norm for functions, and multi-dimensional SBP discretizations
accurately mimic the divergence theorem.Comment: 18 pages, 3 figure
A direct procedure for interpolation on a structured curvilinear two-dimensional grid
A direct procedure is presented for locally bicubic interpolation on a structured, curvilinear, two-dimensional grid. The physical (Cartesian) space is transformed to a computational space in which the grid is uniform and rectangular by a generalized curvilinear coordinate transformation. Required partial derivative information is obtained by finite differences in the computational space. The partial derivatives in physical space are determined by repeated application of the chain rule for partial differentiation. A bilinear transformation is used to analytically transform the individual quadrilateral cells in physical space into unit squares. The interpolation is performed within each unit square using a piecewise bicubic spline
Efficient Tensor-Product Spectral-Element Operators with the Summation-by-Parts Property on Curved Triangles and Tetrahedra
We present an extension of the summation-by-parts (SBP) framework to
tensor-product spectral-element operators in collapsed coordinates. The
proposed approach enables the construction of provably stable discretizations
of arbitrary order which combine the geometric flexibility of unstructured
triangular and tetrahedral meshes with the efficiency of sum-factorization
algorithms. Specifically, a methodology is developed for constructing
triangular and tetrahedral spectral-element operators of any order which
possess the SBP property (i.e. satisfying a discrete analogue of integration by
parts) as well as a tensor-product decomposition. Such operators are then
employed within the context of discontinuous spectral-element methods based on
nodal expansions collocated at the tensor-product quadrature nodes as well as
modal expansions employing Proriol-Koornwinder-Dubiner polynomials, the latter
approach resolving the time step limitation associated with the singularity of
the collapsed coordinate transformation. Energy-stable formulations for
curvilinear meshes are obtained using a skew-symmetric splitting of the metric
terms, and a weight-adjusted approximation is used to efficiently invert the
curvilinear modal mass matrix. The proposed schemes are compared to those using
non-tensorial multidimensional SBP operators, and are found to offer comparable
accuracy to such schemes in the context of smooth linear advection problems on
curved meshes, but at a reduced computational cost for higher polynomial
degrees.Comment: 26 pages, 5 figure
Entropy-split multidimensional summation-by-parts discretization of the Euler and compressible Navier-Stokes equations
High-order Hadamard-form entropy stable multidimensional summation-by-parts
discretizations of the Euler and compressible Navier-Stokes equations are
considerably more expensive than the standard divergence-form discretization.
In search of a more efficient entropy stable scheme, we extend the
entropy-split method for implementation on unstructured grids and investigate
its properties. The main ingredients of the scheme are Harten's entropy
functions, diagonal- summation-by-parts operators with diagonal
norm matrix, and entropy conservative simultaneous approximation terms (SATs).
We show that the scheme is high-order accurate and entropy conservative on
periodic curvilinear unstructured grids for the Euler equations. An entropy
stable matrix-type interface dissipation operator is constructed, which can be
added to the SATs to obtain an entropy stable semi-discretization.
Fully-discrete entropy conservation is achieved using a relaxation Runge-Kutta
method. Entropy stable viscous SATs, applicable to both the Hadamard-form and
entropy-split schemes, are developed for the compressible Navier-Stokes
equations. In the absence of heat fluxes, the entropy-split scheme is entropy
stable for the compressible Navier-Stokes equations. Local conservation in the
vicinity of discontinuities is enforced using an entropy stable hybrid scheme.
Several numerical problems involving both smooth and discontinuous solutions
are investigated to support the theoretical results. Computational cost
comparison studies suggest that the entropy-split scheme offers substantial
efficiency benefits relative to Hadamard-form multidimensional SBP-SAT
discretizations.Comment: 34 pages, 8 figure
A holistic approach for the selection of forensic DNA swabs.
In the present study, we compared the performance of five different ISO 18385 certified forensic swabs for DNA sampling in practice over a time period of five months. Comparisons were made for DNA profiling success rates, measured as the percentage of CODIS (Combined DNA Index System) suitable profiles as well as for practical suitability during sampling at the scene, measured through a survey among collaborators. More than forty members of our crime scene investigation (CSI) unit took part in the test series and provided structured feedback concerning different aspects of swab handling. A total number of 1094 "touch" DNA samples have been subjected to DNA analysis. Swabs performed significantly different in terms of DNA profiling success rates. We also observed significant differences in DNA extraction efficiency between swabs. The evaluation by the collaborators of various aspects of handling differed significantly between swabs. We can assume that a more convenient handling decreases the risk of contamination or sample mislabelling and increases sampling efficiency and staff satisfaction. Our results demonstrate that the selection of disposable sampling devices such as forensic swabs for DNA sampling should be made based on a holistic approach. To be able to select the best performing swab for a given combination of CSI and DNA laboratory procedures, it might not be sufficient to only perform DNA extraction comparisons and trace sampling under controlled laboratory conditions
Quadrature Rules on Triangles and Tetrahedra for Multidimensional Summation-By-Parts Operators
Multidimensional diagonal-norm summation-by-parts (SBP) operators with
collocated volume and facet nodes, known as diagonal- operators,
are attractive for entropy-stable discretizations from an efficiency
standpoint. However, there is a limited number of such operators, and those
currently in existence often have a relatively high node count for a given
polynomial order due to a scarcity of suitable quadrature rules. We present
several new symmetric positive-weight quadrature rules on triangles and
tetrahedra that are suitable for construction of diagonal- SBP
operators. For triangles, quadrature rules of degree one through twenty with
facet nodes that correspond to the Legendre-Gauss-Lobatto (LGL) and
Legendre-Gauss (LG) quadrature rules are derived. For tetrahedra, quadrature
rules of degree one through ten are presented along with the corresponding
facet quadrature rules. All of the quadrature rules are provided in a
supplementary data repository. The quadrature rules are used to construct novel
SBP diagonal- operators, whose accuracy and maximum timestep
restrictions are studied numerically.Comment: 11 pages, 1 figur
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