383 research outputs found
A New Triangular Spectral Element Method I: Implementation and Analysis on a Triangle
This paper serves as our first effort to develop a new triangular spectral
element method (TSEM) on unstructured meshes, using the rectangle-triangle
mapping proposed in the conference note [21]. Here, we provide some new
insights into the originality and distinctive features of the mapping, and show
that this transform only induces a logarithmic singularity, which allows us to
devise a fast, stable and accurate numerical algorithm for its removal.
Consequently, any triangular element can be treated as efficiently as a
quadrilateral element, which affords a great flexibility in handling complex
computational domains. Benefited from the fact that the image of the mapping
includes the polynomial space as a subset, we are able to obtain optimal -
and -estimates of approximation by the proposed basis functions on
triangle. The implementation details and some numerical examples are provided
to validate the efficiency and accuracy of the proposed method. All these will
pave the way for developing an unstructured TSEM based on, e.g., the
hybridizable discontinuous Galerkin formulation
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
Invariant-based approach to symmetry class detection
In this paper, the problem of the identification of the symmetry class of a
given tensor is asked. Contrary to classical approaches which are based on the
spectral properties of the linear operator describing the elasticity, our
setting is based on the invariants of the irreducible tensors appearing in the
harmonic decomposition of the elasticity tensor [Forte-Vianello, 1996]. To that
aim we first introduce a geometrical description of the space of elasticity
tensors. This framework is used to derive invariant-based conditions that
characterize symmetry classes. For low order symmetry classes, such conditions
are given on a triplet of quadratic forms extracted from the harmonic
decomposition of the elasticity tensor , meanwhile for higher-order classes
conditions are provided in terms of elements of , the higher irreducible
space in the decomposition of . Proceeding in such a way some well known
conditions appearing in the Mehrabadi-Cowin theorem for the existence of a
symmetry plane are retrieved, and a set of algebraic relations on polynomial
invariants characterizing the orthotropic, trigonal, tetragonal, transverse
isotropic and cubic symmetry classes are provided. Using a genericity
assumption on the elasticity tensor under study, an algorithm to identify the
symmetry class of a large set of tensors is finally provided.Comment: 32 page
Spectral/hp element methods: recent developments, applications, and perspectives
The spectral/hp element method combines the geometric flexibility of the
classical h-type finite element technique with the desirable numerical
properties of spectral methods, employing high-degree piecewise polynomial
basis functions on coarse finite element-type meshes. The spatial approximation
is based upon orthogonal polynomials, such as Legendre or Chebychev
polynomials, modified to accommodate C0-continuous expansions. Computationally
and theoretically, by increasing the polynomial order p, high-precision
solutions and fast convergence can be obtained and, in particular, under
certain regularity assumptions an exponential reduction in approximation error
between numerical and exact solutions can be achieved. This method has now been
applied in many simulation studies of both fundamental and practical
engineering flows. This paper briefly describes the formulation of the
spectral/hp element method and provides an overview of its application to
computational fluid dynamics. In particular, it focuses on the use the
spectral/hp element method in transitional flows and ocean engineering.
Finally, some of the major challenges to be overcome in order to use the
spectral/hp element method in more complex science and engineering applications
are discussed
Some constants related to numerical ranges
In an attempt to progress towards proving the conjecture the numerical range
W (A) is a 2--spectral set for the matrix A, we propose a study of various
constants. We review some partial results, many problems are still open. We
describe our corresponding numerical tests
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Optimal convergence estimates for the trace of the polynomial L2-projection operator on a simplex
In this paper we study convergence of the L2-projection onto the space of polynomials up to degree p on a simplex in Rd, d >= 2. Optimal error estimates are established in the case of Sobolev regularity and illustrated on several numerical examples. The proof is based on the collapsed coordinate transform and the expansion into various polynomial bases involving Jacobi polynomials and their antiderivatives. The results of the present paper generalize corresponding estimates for cubes in Rd from [P. Houston, C. Schwab, E. Süli, Discontinuous hp-finite element methods for advection-diffusion-reaction problems. SIAM J. Numer. Anal. 39 (2002), no. 6, 2133-2163]
The Medium Amplitude Response of Nonlinear Maxwell-Oldroyd Type Models in Simple Shear
A general framework for Maxwell-Oldroyd type differential constitutive models
is examined, in which an unspecified nonlinear function of the stress and
rate-of-deformation tensors is incorporated into the well-known corotational
version of the Jeffreys model discussed by Oldroyd. For medium amplitude simple
shear deformations, the recently developed mathematical framework of medium
amplitude parallel superposition (MAPS) rheology reveals that this generalized
nonlinear Maxwell model can produce only a limited number of distinct
signatures, which combine linearly in a well-posed basis expansion for the
third order complex viscosity. This basis expansion represents a library of
MAPS signatures for distinct constitutive models that are contained within the
generalized nonlinear Maxwell model. We describe a framework for quantitative
model identification using this basis expansion, and discuss its limitations in
distinguishing distinct nonlinear features of the underlying constitutive
models from medium amplitude shear stress data. The leading order contributions
to the normal stress differences are also considered, revealing that only the
second normal stress difference provides distinct information about the weakly
nonlinear response space of the model. After briefly considering the conditions
for time-strain separability within the generalized nonlinear Maxwell model, we
apply the basis expansion of the third order complex viscosity to derive the
medium amplitude signatures of the model in specific shear deformation
protocols. Finally, we use these signatures for estimation of model parameters
from rheological data obtained by these different deformation protocols,
revealing that three-tone oscillatory shear deformations produce data that is
readily able to distinguish all features of the medium amplitude, simple shear
response space of this generalized class of constitutive models.Comment: 26 pages, 11 figure
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