6,229 research outputs found
A computerized symbolic integration technique for development of triangular and quadrilateral composite shallow-shell finite elements
Computerized symbolic integration was used in conjunction with group-theoretic techniques to obtain analytic expressions for the stiffness, geometric stiffness, consistent mass, and consistent load matrices of composite shallow shell structural elements. The elements are shear flexible and have variable curvature. A stiffness (displacement) formulation was used with the fundamental unknowns consisting of both the displacement and rotation components of the reference surface of the shell. The triangular elements have six and ten nodes; the quadrilateral elements have four and eight nodes and can have internal degrees of freedom associated with displacement modes which vanish along the edges of the element (bubble modes). The stiffness, geometric stiffness, consistent mass, and consistent load coefficients are expressed as linear combinations of integrals (over the element domain) whose integrands are products of shape functions and their derivatives. The evaluation of the elemental matrices is divided into two separate problems - determination of the coefficients in the linear combination and evaluation of the integrals. The integrals are performed symbolically by using the symbolic-and-algebraic-manipulation language MACSYMA. The efficiency of using symbolic integration in the element development is demonstrated by comparing the number of floating-point arithmetic operations required in this approach with those required by a commonly used numerical quadrature technique
Structured backward errors for eigenvalues of linear port-Hamiltonian descriptor systems
When computing the eigenstructure of matrix pencils associated with the
passivity analysis of perturbed port-Hamiltonian descriptor system using a
structured generalized eigenvalue method, one should make sure that the
computed spectrum satisfies the symmetries that corresponds to this structure
and the underlying physical system. We perform a backward error analysis and
show that for matrix pencils associated with port-Hamiltonian descriptor
systems and a given computed eigenstructure with the correct symmetry structure
there always exists a nearby port-Hamiltonian descriptor system with exactly
that eigenstructure. We also derive bounds for how near this system is and show
that the stability radius of the system plays a role in that bound
Numerical implementation of the exact dynamics of free rigid bodies
In this paper the exact analytical solution of the motion of a rigid body
with arbitrary mass distribution is derived in the absence of forces or
torques. The resulting expressions are cast into a form where the dependence of
the motion on initial conditions is explicit and the equations governing the
orientation of the body involve only real numbers. Based on these results, an
efficient method to calculate the location and orientation of the rigid body at
arbitrary times is presented. This implementation can be used to verify the
accuracy of numerical integration schemes for rigid bodies, to serve as a
building block for event-driven discontinuous molecular dynamics simulations of
general rigid bodies, and for constructing symplectic integrators for rigid
body dynamics.Comment: Shortened paper with updated references, 28 pages, 3 figure
Fatal Degeneracy in the Semidefinite Programming Approach to the Decision of Polynomial Inequalities
In order to verify programs or hybrid systems, one often needs to prove that
certain formulas are unsatisfiable. In this paper, we consider conjunctions of
polynomial inequalities over the reals. Classical algorithms for deciding these
not only have high complexity, but also provide no simple proof of
unsatisfiability. Recently, a reduction of this problem to semidefinite
programming and numerical resolution has been proposed. In this article, we
show how this reduction generally produces degenerate problems on which
numerical methods stumble
The tame-wild principle for discriminant relations for number fields
Consider tuples of separable algebras over a common local or global number
field, related to each other by specified resolvent constructions. Under the
assumption that all ramification is tame, simple group-theoretic calculations
give best possible divisibility relations among the discriminants. We show that
for many resolvent constructions, these divisibility relations continue to hold
even in the presence of wild ramification.Comment: 31 pages, 11 figures. Version 2 fixes a normalization error: |G| is
corrected to n in Section 7.5. Version 3 fixes an off-by-one error in Section
6.
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