It is shown that there exists a variational Kohn-Sham density-functional theory, with a minimum principle, for the self-consistent determination of an individual excited-state energy and density. Exact properties of the required functional are ascertained, including a Koopmans theorem. This knowledge allows the employment of an effective potential that gives encouraging numerical results, and also helps to explain the success of a recent perturbation theory and its time-dependent counterpart. 31.15.Ew, 31.50. + w Density-functional theory (DFT) is now in widespread use as an effective approach for ground-state electronic structure calculations. The development of accurate functionals within the popular Kohn-Sham formulation of DFT has enabled us to perform efficient ground-state variational calculations with remarkable accuracy. In Kohn-Sham theory, the simplicity of the three-dimensional electron density is coupled with the use of a relatively small number of orbitals to ensure Fermi statistics, through the use of an auxiliary noninteracting system (see, for instance,There has also been noteworthy progress in excited-state DFT (see, for example, Refs. ). These studies have stimulated us into asking if there exists a variational KohnSham theory for an individual excited state, which is analogous to the ground-state theory, because an affirmative answer implies the possibility that accurate excited-state calculations might be performed routinely, in a manner comparable to today's ground-state calculations. Accordingly, it is our purpose to show that there does indeed exist such a variational Kohn-Sham theory, with a minimum principle, for an individual excited state. In our proof, the necessary universal functional is identified and several of its properties are ascertained for the purpose of approximation. This enables us to actually carry out illustrative self-consistent calculations, and encouraging results are obtained for the systems studied.Consider the Hamiltonian of interestĤ y :whereT is the kinetic energy operator,V ee is the electron-electron repulsion operator, and y͑ r͒ is the localmultiplicative attractive potential of interest. Assume we want the energy and density of the kth state ofĤ y . (In this Letter, all interacting and noninteracting states shall be assumed nondegenerate to facilitate the presentation.) For this purpose we start by generalizing earlier excited-state functionals [12,13] and define the universal bywhere both r and r 0 are arbitrary electron densities. In Eq. (2), it is understood that each C is orthogonal to the first k 2 1 states of that Hamiltonian,Ĥ y 0 T 1V ee 1 P N i1 y 0 ͑ r i ͒, for which r 0 is the ground-state density. It follows from the definition of F͓r, r 0 ͔ that E k , the energy of the kth state ofĤ y , is given bywhere r 0 is the ground-state density ofĤ y and r k is the density of its kth state. Analogous with the constrainedsearch proof of the ground-state Hohenberg-Kohn variational theorem, Eq. (3) is true becausewhere the C's are understood to be restri...

To cite this version:Arnaud Lazarus, J.T. Miller, P.M. Reis. Continuation of equilibria and stability of slender elastic rods using an asymptotic numerical method.We present a theoretical and numerical framework to compute bifurcations of equilibria and stability of slender elastic rods. The 3D kinematics of the rod is treated in a geometrically exact way by parameterizing the position of the centerline and making use of quaternions to represent the orientation of the material frame. The equilibrium equations and the stability of their solutions are derived from the mechanical energy which takes into account the contributions due to internal moments (bending and twist), external forces and torques. Our use of quaternions allows for the equilibrium equations to be written in a quadratic form and solved efficiently with an asymptotic numerical continuation method. This finite element perturbation method gives interactive access to semi-analytical equilibrium branches, in contrast with the individual solution points obtained from classical minimization or predictor-corrector techniques. By way of example, we apply our numerics to address the specific problem of a naturally curved and heavy rod under extreme twisting and perform a detailed comparison against our own precision model experiments of this system. Excellent quantitative agreement is found between experiments and simulations for the underlying 3D buckling instabilities and the characterization of the resulting complex configurations. We believe that our framework is a powerful alternative to other methods for the computation of nonlinear equilibrium 3D shapes of rods in practical scenarios.1 range from the supercoiling of DNA (Coleman and Swigon, 2000;Marko and Neukirch 2012), self-assembly of rod-coil block copolymers (Wang et al., 2012), design of nano-electromechanical resonators (Lazarus et al., 2010b(Lazarus et al., , 2010a, development of stretchable electronics (Sun et al., 2006), computed animation of hairs (Bertails et al., 2006) and coiled tubing operations in the oil-gas industries (Wicks et al., 2008).An ongoing challenge in addressing these various problems involves the capability to numerically capture their intrinsic geometric nonlinearities in a predictive and efficient way. These nonlinear kinematic effects arise from the large displacements and rotations of the slender structure, even if its material properties remain linear throughout the process (Audoly and Pomeau, 2010). As a slender elastic rod is progressively deformed, the nonlinearities of the underlying equilibrium equations become increasingly stronger leading to higher densities in the landscape of possible solutions for a particular set of control parameters. When multiple stable states coexist, classic step-by-step algorithms such as Newton-Raphson methods (Crisfield, 1991) or standard minimization techniques (Luenberger, 1973) are often inappropriate since, depending on the initial guess, they may not converge toward the desired solution, or any solution. Addressing these com...

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We present results from an experimental and numerical investigation on the compression, and consequent buckling, of a slender rod constrained inside a horizontal cylinder. An experimental model system is developed to systematically study the sequence of instabilities from straight-to-sinusoidal and sinusoidal-to-helical configurations. We quantify the associated buckling loads as a function of the radial clearance between the rod and cylindrical constraint. These results are compared to existing theoretical predictions. While good agreement is obtained for large values of the radial clearance, significant deviations are found when the geometric imperfections of the setup are comparable to the radial clearance. Due to this imperfection sensitivity, the critical buckling loads can be reduced significantly by up to a factor of three. The findings from this model system can be applied to practical applications across a range of length scales due to the geometric (rather than material) nonlinearities involved in the deformations of rods.

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