scholarly journals Bimodal optimization with constraints: Critical value of the constraint and post-critical configurations

2011 ◽  
Vol 38 (2) ◽  
pp. 107-124
Author(s):  
Teodor Atanackovic ◽  
Alexander Seyraniany
Keyword(s):  
Total Energy ◽  
Critical Value ◽  
Optimal Shape ◽  
Cross Sectional Area ◽  
Point Of View ◽  
Sectional Area ◽  
Cross Sectional ◽  
Critical Configurations ◽  
Bimodal Optimization ◽  
Elastic Column

By using a method based on Pontryagin?s principle, formulated in [13], and [14] we study optimal shape of an elastic column with constraints on the minimal value of the cross-sectional area. We determine the critical value of the minimal cross-sectional area separating bi from unimodal optimization. Also we study the post-critical shape of optimally shaped rod and find the preferred configuration of the bifurcating solutions from the point of view of minimal total energy.

APPLICATION OF PONTRYAGIN'S PRINCIPLE TO BIMODAL OPTIMIZATION OF NANO RODS

2012 ◽  
Vol 12 (03) ◽  
pp. 1250012 ◽  
Author(s):  
TEODOR M. ATANACKOVIC ◽  
BRANISLAVA N. NOVAKOVIC ◽  
ZORA VRCELJ
Keyword(s):  
Maximum Principle ◽  
Characteristic Length ◽  
Optimization Procedure ◽  
Optimal Shape ◽  
Cross Sectional Area ◽  
Sectional Area ◽  
Elastic Rod ◽  
Cross Sectional ◽  
Bimodal Optimization ◽  
Nano Rods

By using the Pontryagin's maximum principle, we determine optimal shape of a nonlocal elastic rod clamped at both ends. In the optimization procedure, we imposed restriction on the minimal value of the cross-sectional area. We showed that the optimization may be both unimodal and bimodal depending on the value of the restrictions and the value of characteristic length. Several concrete examples are treated in detail and the increase in buckling capacity is determined.

Effect of Weld Defects on Tensile Properties of Lightweight Materials and Correlations With Phased Array Ultrasonic Nondestructive Evaluation

2014 ◽  
Author(s):  
Mohammad W. Dewan ◽  
M. A. Wahab ◽  
Ayman M. Okeil
Keyword(s):  
Tensile Strength ◽  
Aluminum Alloy ◽  
Tensile Properties ◽  
Phased Array ◽  
Critical Value ◽  
Area Ratio ◽  
Cross Sectional Area ◽  
Sectional Area ◽  
Cross Sectional ◽  
Strength And Toughness

Fusion welding of Aluminum and its alloys is a great challenge for the structural integrity of lightweight material structures. One of the major shortcomings of Aluminum alloy welding is the inherent existence of defects in the welded area. In the current study, tests have been conducted on tungsten inert gas (TIG) welded AA6061-T651 aluminum alloy to determine the effects of defect sizes and its distribution on fracture strength. The information will be used to establish weld acceptance/rejection criteria. After welding, all specimens were non-destructively inspected with phased array ultrasonic and measured the projected area of the defects. Tensile testing was performed on inspected specimens containing different weld defects: such as, porosity, lack of fusion, and incomplete penetration. Tensile tested samples were cut along the cross section and inspected with Optical Microscope (OM) to measure actual defect sizes. Tensile properties were correlated with phased array ultrasonic testing (PAUT) results and through microscopic evaluations. Generally, good agreement was found between PAUT and microscopic defect sizing. The tensile strength and toughness decreased with the increase of defect sizes. Small voids (area ratio <0.04) does not have significant effect on the reduction of tensile strength and toughness values. Once defective “area ratio (cross sectional area of the defect) / (total specimen cross sectional area)” reached a certain critical value (say, 0.05), both strength and toughness values decline sharply. After that critical value both the tensile strength and toughness values decreases linearly with the increase of defect area ratio.

Optimal Shape of a Rotating Rod With Unsymmetrical Boundary Conditions

2007 ◽  
Vol 74 (6) ◽  
pp. 1234-1238 ◽  
Author(s):  
Teodor M. Atanackovic
Keyword(s):  
Maximum Principle ◽  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Optimal Shape ◽  
Elastic Rods ◽  
Sectional Area ◽  
Governing Equations ◽  
Cross Sectional ◽  
Bimodal Optimization ◽  
Multiplicity Of An Eigenvalue

Governing equations of a compressed rotating rod with clamped–elastically clamped (hinged with a torsional spring) boundary conditions is derived. It is shown that the multiplicity of an eigenvalue of this system can be at most equal to two. The optimality conditions, via Pontryagin’s maximum principle, are derived in the case of bimodal optimization. When these conditions are used the problem of determining the optimal cross-sectional area function is reduced to the solution of a nonlinear boundary value problem. The problem treated here generalizes our earlier results presented in Atanackovic, 1997, Stability Theory of Elastic Rods, World Scientific, River Edge, NJ. The optimal shape of a rod is determined by numerical integration for several values of parameters.

scholarly journals Optimal shape of a column with clamped-elastically supported ends positioned on elastic foundation

2015 ◽  
Vol 42 (3) ◽  
pp. 191-200 ◽  
Author(s):  
Branislava Novakovic
Keyword(s):  
Optimality Conditions ◽  
Elastic Foundation ◽  
Compressive Force ◽  
First Integral ◽  
Optimal Shape ◽  
Cross Sectional ◽  
Bimodal Optimization ◽  
Elastic Column ◽  
Elastically Supported ◽  
Linear Boundary Value Problem

We determine optimal shape of an elastic column positioned on elastic foundation of Winkler type. The Euler-Bernoulli model of beam is considered. The column is loaded by a compressive force and has one clamped end and the other elastically supported end. In deriving the optimality conditions, the Pontryagin?s principle was used. The optimality conditions for the case of bimodal optimization are derived. Optimal cross-sectional area is obtained from the solution of a non-linear boundary value problem. A first integral (Hamiltonian) is used to monitor accuracy of integration. This system is solved by using standard Math CAD procedure. New numerical results are obtained.

Resilient Energy

1943 ◽  
Vol 16 (3) ◽  
pp. 591-608
Author(s):  
E. C. B. Bott
Keyword(s):  
Tensile Strength ◽  
Finite Differences ◽  
Practical Method ◽  
Cross Sectional Area ◽  
Point Of View ◽  
Sectional Area ◽  
Cross Sectional ◽  
Vulcanized Rubber ◽  
Reinforcing Agent ◽  
Strength Based

Abstract 1. The tensile strength of vulcanized rubber may be expressed in terms of its elongation by means of the calculus of finite differences. 2. This expression for tensile strength, based on the theoretical cross-sectional area, gives an expression for the tensile strength based on the original cross-sectional area when the former quantity is divided by the factor (E + 1), E being the elongation. 3. The expression for tensile strength based on the original cross-sectional area is integrated with respect to the elongation to give the resilient energy. 4. The trapezoidal rule has proved itself to be superior to the calculus of finite differences as a practical method of obtaining the resilient energy. 5. The total resilient energies are plotted on graphs against the percentage by volume of reinforcing agent or filler. Tangents drawn at any desired point corresponding to a certain percentage of filler give values for the partial resilient energies of base mix and of filler by the method of tangent intercepts. 6. The expressions for tensile strength are composed of one, two or three functions ; the number of functions is, in general, inversely proportional to the percentage of the filler in the vulcanizate. 7. The expressions for tensile strength and for resilient energies have no significance regarding the structure of vulcanized rubber; they have been evolved from the point of view of usefulness for evaluating compounds. 8. The values of the partial resilient energies of base mix and of filler obtained by the method of tangent intercepts have no physical meaning; they are a means of calculating the total resilient energy of a sample of vulcanized rubber.

scholarly journals On the optimal shape of a compressed column

2010 ◽  
Vol 37 (1) ◽  
pp. 37-48
Author(s):  
Teodor Atanackovic ◽  
Branislava Novakovic ◽  
Emina Basara
Keyword(s):  
Numerical Experiments ◽  
A Priori Estimates ◽  
A Priori ◽  
Optimal Shape ◽  
Cross Sectional Area ◽  
Sectional Area ◽  
Cross Sectional ◽  
Lagrange Problem ◽  
Priori Estimates ◽  
The Cross

A new numerical solution to the Lagrange problem is presented. The solution is compared with a priori estimates obtained recently. Also we derive a new relation that shows that the cross-sectional area at the middle of the optimally shaped column is larger than the cross-sectional area at the ends. Our numerical experiments confirm that conclusion.

scholarly journals A New Approach on Vibrating Horns Design

2017 ◽  
Vol 2017 ◽  
pp. 1-12 ◽  
Author(s):  
Maria Violeta Guiman ◽  
Ioan Călin Roșca
Keyword(s):  
Optimization Design ◽  
Nodal Point ◽  
Optimization Method ◽  
Cross Sectional Area ◽  
Point Of View ◽  
Sectional Area ◽  
Cross Sectional ◽  
New Approach ◽  
Frequency Maximum ◽  
Predicted Values

An optimization method of the vibrating horns is presented considering the smallest action principle and the attached cutting tool mass. The model is based on Webster’s wave propagation equation and as an objective function the minimization of the volume in structural equilibrium conditions was considered. The considered input parameters were working frequency, maximum cross-sectional area, magnification coefficient, and the attached mass. At the end of the study, a new shape function of the horn’s cross section is obtained. The particularity of the new obtained shape is given by the nodal point position that is the same with the position of the maximum cross-sectional area. The obtained horn was analyzed from the modal point of view using theoretical and experimental methods. As theoretical methods, both the state-space method and the finite element method were used. An experimental setup for frequency response function determination was developed using a random input signal. The verification of the magnitude value was done considering a harmonic steady-state signal. The recorded values were compared with the predicted values. The numerical simulations and tests support the validity of the assumptions used in the horns optimization design.

Fascicular Topography of the Suprascapular Nerve in the C5 Root and Upper Trunk of the Brachial Plexus: A Microanatomic Study From a Nerve Surgeon's Perspective

2010 ◽  
Vol 67 (suppl_2) ◽  
pp. ons402-ons406 ◽  
Author(s):  
Mario G. Siqueira ◽  
Luciano H.L. Foroni ◽  
Roberto S. Martins ◽  
Gerson Chadi ◽  
Martijn J.A. Malessy
Keyword(s):  
Brachial Plexus ◽  
Sural Nerve ◽  
Specific Area ◽  
Suprascapular Nerve ◽  
Nerve Graft ◽  
Cross Sectional Area ◽  
Point Of View ◽  
Sectional Area ◽  
Cross Sectional ◽  
Sural Nerve Graft

ABSTRACT BACKGROUND: In patients with supraclavicular injuries of the brachial plexus, the suprascapular nerve (SSN) is frequently reconstructed with a sural nerve graft coapted to C5. As the C5 cross-sectional diameter exceeds the graft diameter, inadequate positioning of the graft is possible. OBJECTIVE: To identify a specific area within the C5 proximal stump that contains the SSN axons and to determine how this area could be localized by the nerve surgeon, we conducted a microanatomic study of the intraplexal topography of the SSN. METHODS: The right-sided C5 and C6 roots, the upper trunk with its divisions, and the SSN of 20 adult nonfixed cadavers were removed and fixed. The position and area occupied by the SSN fibers inside C5 were assessed and registered under magnification. RESULTS: The SSN was monofascicular in all specimens and derived its fibers mainly from C5. Small contributions from C6 were found in 12 specimens (60%). The mean transverse area of C5 occupied by SSN fibers was 28.23%. In 16 specimens (80%), the SSN fibers were localized in the ventral (mainly the rostroventral) quadrants of C5, a cross-sectional area between 9 o'clock and 3 o'clock from the surgeon's intraoperative perspective. CONCLUSION: In reconstruction of the SSN with a sural nerve graft, coaptation should be performed in the rostroventral quadrant of C5 cross-sectional area (between 9 and 12 o'clock from the nerve surgeon's point of view in a right-sided brachial plexus exploration). This will minimize axonal misrouting and may improve outcome.

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