Multiple solutions to non-convex variational problems with implications for phase transitions and numerical computation

Non-convex variational/boundary-value problems are studied usinga modified version of the Ericksen bar model in nonlinear elasticity.The strain-energy function is a general fourth-order polynomialin a suitable measure of strain that provides a convenient modelfor the study of, for example, phase transitions. On the basisof a canonical duality theory, the nonlinear differential equationfor the non-convex, non-homogeneous variational problem, herewith either mixed or Dirichlet boundary conditions, is convertedinto an algebraic equation, which can, in principle, be solvedto obtain a complete set of solutions. It should be emphasizedthat one important outcome of the theory is the identificationand characterization of the local energy extrema and the globalenergy minimizer. For the soft loading device criteria for theexistence, uniqueness, smoothness and multiplicity of solutionsare presented and discussed. The iterative finite-differencemethod (FDM) is used to illustrate the difficulty of capturingnon-smooth solutions with traditional FDMs. The results illustratethe important fact that smooth analytic or numerical solutionsof a nonlinear mixed boundary-value problem might not be minimizersof the associated potential variational problem. From a ‘dual’perspective, the convergence (or non-convergence) of the FDMis explained and numerical examples are provided.     

Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem

The pure azimuthal shear problem for a circular cylindrical tube of nonlinearly elastic material, both isotropic and anisotropic, is examined on the basis of a complementary energy principle. For particular choices of strain-energy function, one convex and one non-convex, closed-form solutions are obtained for this mixed boundary-value problem, for which the governing differential equation can be converted into an algebraic equation. The results for the non-convex strain energy function provide an illustration of a situation in which smooth analytic solutions of a nonlinear boundary-value problem are not global minimizers of the energy in the variational statement of the problem. Both the global minimizer and the local extrema are identified and the results are illustrated for particular values of the material parameters.      

Chalcogenide-halides of niobium (V). 1. Gas-phase structures of NbOBr(3), NbSBr(3), and NbSCl(3). 2. Matrix infrared spectra and vibrational force fields of NbOBr(3), NbSBr(3), NbSCl(3), and NbOCl(3)

The molecular structures of NbOBr(3), NbSCl(3), and NbSBr(3) have been determined by gas-phase electron diffraction (GED) at nozzle-tip temperatures of 250 degrees C, taking into account the possible presence of NbOCl(3) as a contaminant in the NbSCl(3) sample and NbOBr(3) in the NbSBr(3) sample. The experimental data are consistent with trigonal-pyramidal molecules having C(3)(v)() symmetry. Infrared spectra of molecules trapped in argon or nitrogen matrices were recorded and exhibit the characteristic fundamental stretching modes for C(3)(v)() species. Well resolved isotopic fine structure ((35)Cl and (37)Cl) was observed for NbSCl(3), and for NbOCl(3) which occurred as an impurity in the NbSCl(3) spectra. Quantum mechanical calculations of the structures and vibrational frequencies of the four YNbX(3) molecules (Y = O, S; X = Cl, Br) were carried out at several levels of theory, most importantly B3LYP DFT with either the Stuttgart RSC ECP or Hay-Wadt (n + 1) ECP VDZ basis set for Nb and the 6-311G basis set for the nonmetal atoms. Theoretical values for the bond lengths are 0.01-0.04 A longer than the experimental ones of type r(a), in accord with general experience, but the bond angles with theoretical minus experimental differences of only 1.0-1.5 degrees are notably accurate. Symmetrized force fields were also calculated. The experimental bond lengths (r(g)/A) and angles ( 90 degree angle (alpha)()/deg) with estimated 2sigma uncertainties from GED are as follows. NbOBr(3): r(Nb=O) = 1.694(7), r(Nb-Br) = 2.429(2), 90 degree angle (O=Nb-Br) = 107.3(5), 90 degree angle (Br-Nb-Br) = 111.5(5). NbSBr(3): r(Nb=S) = 2.134(10), r(Nb-Br) = 2.408(4), 90 degree angle (S=Nb-Br) = 106.6(7), 90 degree angle (Br-Nb-Br) = 112.2(6). NbSCl(3): r(Nb=S) = 2.120(10),r(Nb-Cl) = 2.271(6), 90 degree angle (S=Nb-Cl) = 107.8(12), 90 degree angle (Cl-Nb-Cl) = 111.1(11).     

Nonlinear analysis of flow pulses and shock waves in arteries

Zusammenfassung Druck- und Strompulse mit grosser Amplitude erzeugt in der Aorta und in andern Hauptarterien des Hundes werden theoretisch berechnet für vorgeschriebene Ausströmungspulse von der linken Herzkammer und für gegebene physikalische und geometrische Eigenschaften des Kreislauf-systems. Der Blutausfluss durch die Äste und Verzweigungen der uns interessierenden arteriellen Leitung ist durch ein kontinuierlich verteiltes, von Druck und Ort abhängiges Ausflussmodell nachgeahmt. Am herzfernen Ende der Leitung ist als Randbedingung entweder der periphere Widerstand oder ein konstanter druck vorgeschrieben. Die Geometrie der Leitung ist durch ihren kreisförmingen Querschnitt und den mit Herzdistanz exponentiell abnehmenden Radius definiert. Die elastischen Eigenschaften der Gefässwand sind durch die von Ort und Druck abhängige Geschwindigkeit kleiner Druckwellen gegeben. Durch Integration der Beziehung zwischen Wellen-geschwindigkeit und Querschnittsdehnung ist damit auch der Querschnitt als Funktion des Druckes und der Herzdistanz vorgeschrieben. Die nichtlinearen Gleichungen für eindimensionale Strömung einer inkompressibler Flüssigkeit werden mit Hilfe der Charakteristikenmethode integriert für Kreislaufparameterwerte die einem hypothetischen Hund von 30 kg Gewicht entsprechen. Das verwendete mathematische Modell für die arterielle Leitung wiedergibt manche der bekannten Eigenschaften des vom Herzen erzeugten Pulses, einschliesslich die Klappenincisur, das Ansteigen und Abfallen der Höhe des systolischen Druckgipfels mit wachsender Entfernung vom Herzen. Während der Fortpflanzung der Pulswelle zeichnet sich eine zunehmende Steilheit der Wellenfront ab, die jedoch nicht merkbar ist wenn man die grundlegenden Gleichungen linearisiert. Die numerischen Ergebnisse weisen darauf hin, dass die sekundäre (dicrotic) Welle durch Reflexionen erzeugt wird, und als solche von der Verjüngung des Querschnittes und vom Blutausfluss abhängt.

---

Notation A cross-sectional area of artery at the reference intraluminal pressurep ₀ - c local wave speed - c ₀,c ₁ parameters in wave speed expression - c _L local wave speed for linearized analysis - C, C designations for general curves inz, t plane - D ₀ diameter at aortic valve whenp=p ₀ - D _t diameter at distal end of artery whenp=p ₀ - E circumferential Young's modulus - f axial frictional force per unit mass of fluid - h wall thickness - L distance from aortic valve to distal end of artery - n parameter in wave speed expression - p intraluminal pressure - calculated mean pressure - p ₀ reference pressure - p _c capillary pressure - p _L pressure at distal end of artery - q S v=local volume flow rate - q ₀ q ₀(t)=volume flow rate ejected by heart - r radial coordinate - R _e Reynolds number for steady flow - R _L peripheral resistance - s curvilinear coordinate - S cross-sectional area of artery - S _L cross-sectional area for linearized analysis - t time - v axial fluid velocity averaged over cross section - z axial distance coordinate - z ^* distance from aortic valve to femoral artery - exponent in cross-sectional area expression - outflow parameter - undetermined multiplier - blood viscosity coefficient - blood density - outflow function simulating effect of side branches