18,943 research outputs found
An isogeometric finite element formulation for phase transitions on deforming surfaces
This paper presents a general theory and isogeometric finite element
implementation for studying mass conserving phase transitions on deforming
surfaces. The mathematical problem is governed by two coupled fourth-order
nonlinear partial differential equations (PDEs) that live on an evolving
two-dimensional manifold. For the phase transitions, the PDE is the
Cahn-Hilliard equation for curved surfaces, which can be derived from surface
mass balance in the framework of irreversible thermodynamics. For the surface
deformation, the PDE is the (vector-valued) Kirchhoff-Love thin shell equation.
Both PDEs can be efficiently discretized using -continuous interpolations
without derivative degrees-of-freedom (dofs). Structured NURBS and unstructured
spline spaces with pointwise -continuity are utilized for these
interpolations. The resulting finite element formulation is discretized in time
by the generalized- scheme with adaptive time-stepping, and it is fully
linearized within a monolithic Newton-Raphson approach. A curvilinear surface
parameterization is used throughout the formulation to admit general surface
shapes and deformations. The behavior of the coupled system is illustrated by
several numerical examples exhibiting phase transitions on deforming spheres,
tori and double-tori.Comment: fixed typos, extended literature review, added clarifying notes to
the text, added supplementary movie file
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
A hysteretic multiscale formulation for nonlinear dynamic analysis of composite materials
This article has been made available through the Brunel Open Access Publishing Fund.A new multiscale finite element formulation
is presented for nonlinear dynamic analysis of heterogeneous
structures. The proposed multiscale approach utilizes
the hysteretic finite element method to model the microstructure.
Using the proposed computational scheme, the micro-basis functions, that are used to map the microdisplacement components to the coarse mesh, are only evaluated once and remain constant throughout the analysis procedure. This is accomplished by treating inelasticity at the micro-elemental level through properly defined hysteretic evolution equations. Two types of imposed boundary conditions are considered for the derivation of the multiscale basis functions, namely the linear and periodic boundary conditions. The validity of the proposed formulation as well as its computational efficiency are verified through illustrative numerical experiments
Statistical Lyapunov theory based on bifurcation analysis of energy cascade in isotropic homogeneous turbulence: a physical -- mathematical review
This work presents a review of previous articles dealing with an original
turbulence theory proposed by the author, and provides new theoretical insights
into some related issues. The new theoretical procedures and methodological
approaches confirm and corroborate the previous results. These articles study
the regime of homogeneous isotropic turbulence for incompressible fluids and
propose theoretical approaches based on a specific Lyapunov theory for
determining the closures of the von K\'arm\'an-Howarth and Corrsin equations,
and the statistics of velocity and temperature difference. Furthermore, novel
theoretical issues are here presented among which we can mention the following
ones. The bifurcation rate of the velocity gradient, calculated along fluid
particles trajectories, is shown to be much larger than the corresponding
maximal Lyapunov exponent. On that basis, an interpretation of the energy
cascade phenomenon is given and the statistics of finite time Lyapunov exponent
of the velocity gradient is shown to be represented by normal distribution
functions. Next, the self--similarity produced by the proposed closures is
analyzed, and a proper bifurcation analysis of the closed von
K\'arm\'an--Howarth equation is performed. This latter investigates the route
from developed turbulence toward the non--chaotic regimes, leading to an
estimate of the critical Taylor scale Reynolds number. A proper statistical
decomposition based on extended distribution functions and on the
Navier--Stokes equations is presented, which leads to the statistics of
velocity and temperature difference.Comment: physical--mathematical review of previous works and new theoretical
insights into some relates issue
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