Adaptive Finite Element Method for Solving Poisson Partial Differential Equation

Total Page:16

File Type:pdf, Size:1020Kb

Adaptive Finite Element Method for Solving Poisson Partial Differential Equation Journal of Nepal Mathematical Society (JNMS), Vol. 4, Issue 1(2021); K.C. Gokul, R. P. Dulal Adaptive Finite Element Method for Solving Poisson Partial Differential Equation Gokul K C1, Ram Prasad Dulal1 1 Department of Mathematics, School of Science, Kathmandu University, Dhulikhel, Nepal Correspondence to: Ram Prasad Dulal, Email: [email protected] Abstract: Poisson equation is an elliptic partial differential equation, a generalization of Laplace equation. Finite element method is a widely used method for numerically solving partial differential equations. Adap- tive finite element method distributes more mesh nodes around the area where singularity of the solution happens. In this paper, Poisson equation is solved using finite element method in a rectangular domain with Dirichlet and Neumann boundary conditions. Posteriori error analysis is used to mark the refinement area of the mesh. Dorfler adaptive algorithm is used to refine the marked mesh. The obtained results are compared with exact solutions and displayed graphically. Keywords: Poisson equation, Finite element method, Posteriori error analysis, Adaptive mesh refinement DOI: https://doi.org/10.3126/jnms.v4i1.37107 1 Introduction Differential equations arise in many areas of applications such as in science and engineering. Most of the problems in almost all the scientific and engineering areas are modeled using partial differential equations (PDEs). Partial differential equation arise when a dependent variable depends on two or more independent variables. One of the widely known method for solving PDEs is finite element method (FEM). The Finite element method is a numerical analysis technique for obtaining approximate solutions to a wide variety of engineering problems. A finite element model of a problem gives a piecewise approximation to the governing equations. The basic premise of the FEM is that a solution region can be analytically modeled or approximated by replacing it with an assemblage of discrete elements[5]. 1.1 Poisson partial differential equation Poisson equation is a partial differential equation, named after Simeon Denis Poisson. The Poisson equation is derived from the following general elliptic partial differential equation[8]: @ @u @u @ @u @u − a + a − a + a + a u − f = 0 (1) @x 11 @x 12 @y @y 21 @x 22 @y 00 For particular case, setting a11 = a22 = k(x; y) = k (constant) and a21 = a21 = a00 = 0, the equation reduces to: @2u @2u −k + = f(x; y) @x2 @y2 −kr2u = f ) −k∆u = f (2) 2 @2 @2 where, ∆ = r = @x2 + @y2 is called Laplacian operator. The equation (2) is the simplest form of elliptic PDE called Poisson PDE consisting dependent variable u and independent variables x and y. The function f(x; y) is called source function. 1 Adaptive Finite Element Method for Solving Poisson Partial Differential Equation 1.2 Adaptive finite element method (AFEM) An adaptive finite element method (AFEM) was developed in the early 1980s. The basic concept of adaptivity developed in the FEM is that, when a physical problem is analyzed using finite elements, there exists some discretization errors caused owing to the use of the finite element model. These errors are calculated in order to assess the accuracy of the solution obtained. If the errors are large, the finite element model is refined through reducing the size of elements or increasing the order of interpolation functions. The new model is re-analyzed and the errors in the new model are re-calculated. This procedure is continued until the calculated errors fall below the specified permissible values. The key features in AFEM are the estimation of discretization errors and the refinement of finite element models[7]. It minimizes the errors by the concept of posteriori error analysis and using adaptive algorithm. It can be shown in the following diagram[7]: Figure 1: Adaptive finite element process 2 Methodology 2.1 Weak form of poisson equation In the development of the weak form, we consider a typical triangular element Ωe of the finite element mesh and develop the finite element model for the Poisson PDE (2). Multiplying (2) by an arbitrary test 1 function w = w(x; y) 2 H0 (Ωe), which is assumed to be differentiable with respect to x and y, we get weak form of (2) as[4]: Z @w @u @w @u Z I k + k dxdy = wfdxdy + wqnds (3) Ωe @x @x @y @y Ωe Γe The equation (3) can be written as: Be(w; u) = Le(w) (4) where, Z @w @u @w @u k + k dxdy = Be(w; u) Ωe @x @x @y @y Z I e wfdxdy + wqnds = L (w) Ωe Γe 2.2 Finite element solution process 2.2.1 Interpolation functions The governing equation (2) is equivalent to the weak form of equation (4) over the domain Ωe and also contains natural boundary conditions. The weak form requires the approximation chosen for u should be at least linear in both x and y. There are no terms that are identically zero. Since the primary variable is a function itself, the Lagrange family of interpolation functions is admissible. Let u is the approximated solution over a typical element Ωe defined by the expression[8]: n e X e e u = u(x; y) ≈ uh(x; y) = uj j (x; y): (5) j=1 2 Journal of Nepal Mathematical Society (JNMS), Vol. 4, Issue 1(2021); K.C. Gokul, R. P. Dulal e e th e Where, uj is the value of uh at the j node (xj; yj) of the element and j is the Lagrange interpolation function with the property that: e i (xj; yj) = δij; i = 1; 2; :::; n (6) Putting the approximated value of (5) in (4), we get: n Z e e Z I X @w @ j @w @ j k + k dxdy ue = wfdxdy + wq ds (7) @x @x @y @y j n j=1 Ωe Ωe Γe This equation must hold for every admissible choice of weight function w. Since we need n independent e e e e algebraic equations to solve for n unknowns (u1; u2; u3; :::; un), we choose n linearly independent functions e e e e for w. The functions of w are w = ( 1; 2; 3; :::; n). Hence we write: e e e e e u = u1 u2 u3 : : : un e e e e T w = 1 2 3 : : : n = This particular choice of weight function is a natural one when the weight function is viewed as a virtual Pn variation of the dependent unknowns (w = @u = i=1 @ui i) and the resulting finite element model is called weak form finite element model[8]. e e e e For each choice of w, we obtain an algebraic relation among (u1; u2; u3; :::; un). We label the algebraic equation resulting from substituting successively the following in (7): e e e e w = 1; w = 2; w = 3; :::; w = n: to get n algebraic equations. Hence the general ith form of the system can be written as: n Z e e e e Z I X @ @ j @ @ j i k + i k dxdy ue = f edxdy + q eds where @x @x @y @y j i n i j=1 Ωe Ωe Γe Z e e e e Z I e @ i @ j @ i @ j e e e e Kij = k + k dxdy; fi = f i dxdy; Qi = qn i ds Ωe @x @x @y @y Ωe Γe Which reduces to: n X e e e e Kijuj = fi + Qi ; i = 1; 2; 3; :::; n (8) j=1 e e e e e The term Kij is a bilinear form and is denoted as Kij = B ( i ; j ). Equation 8 abbreviated in vector form as: Keue = f e + Qe (9) The matrix Ke is called the coefficient matrix and the vector f e is called the source vector. The weak form e and the finite element matrices in (9) shows that i must be at least linear functions of x and y. The complete linear polynomial in x and y in Ωe is of the form: e e e e uh(x; y) = c1 + c2x + c3y (10) e where, ci 's are constants for i = 1; 2; 3: The set f1; x; yg is linearly independent and complete. The e e equation (10) defines a unique plane for fixed ci . In particular, uh(x; y) is uniquely defined on a triangle. Let (x1; y1); (x2; y2) and (x3; y3) be the vertices of the triangle such that: e e e e e e uh(x1; y1) = u1; uh(x2; y2) = u2; uh(x3; y3) = u3: (11) e e The constants ci ,(i = 1; 2; 3) in (10) can be expressed in terms of three nodal values ui (i = 1; 2; 3). The particular nodes of the triangle can be expressed as: e e e e e uh(x1; y1) ≡ u1 = c1 + c2x1 + c3y1 e e e e e uh(x2; y2) ≡ u2 = c1 + c2x2 + c3y2 (12) e e e e e uh(x3; y3) ≡ u3 = c1 + c2x3 + c3y3: 3 Adaptive Finite Element Method for Solving Poisson Partial Differential Equation The matrix form of this system of equations is: 2 3 2 e3 2 e3 1 x1 y1 c1 u1 e e 41 x2 y25 4c25 = 4u25 (13) e e 1 x3 y3 c3 u3 where, A is coefficient matrix, C is constant matrix and ue is solution matrix such that: 2 3 2 e3 2 e3 1 x1 y1 c1 u1 e e e A = 41 x2 y25 ; C = 4c25 ; u = 4u25 : e e 1 x3 y3 c3 u3 The inverse matrix of A is given by: 2α α α 3 1 1 2 3 A−1 = : β β β (14) jAj 4 1 2 3 5 γ1 γ2 γ3 where αi; βi; γi are constants depend only on the global coordinates of element nodes (xi; yi) s.t.: αi = xjyk − xkyj; βi = yj − yk; γi = xk − xj (15) for i 6= j 6= k and i; j; k permute in a natural order.
Recommended publications
  • Leonhard Euler: His Life, the Man, and His Works∗
    SIAM REVIEW c 2008 Walter Gautschi Vol. 50, No. 1, pp. 3–33 Leonhard Euler: His Life, the Man, and His Works∗ Walter Gautschi† Abstract. On the occasion of the 300th anniversary (on April 15, 2007) of Euler’s birth, an attempt is made to bring Euler’s genius to the attention of a broad segment of the educated public. The three stations of his life—Basel, St. Petersburg, andBerlin—are sketchedandthe principal works identified in more or less chronological order. To convey a flavor of his work andits impact on modernscience, a few of Euler’s memorable contributions are selected anddiscussedinmore detail. Remarks on Euler’s personality, intellect, andcraftsmanship roundout the presentation. Key words. LeonhardEuler, sketch of Euler’s life, works, andpersonality AMS subject classification. 01A50 DOI. 10.1137/070702710 Seh ich die Werke der Meister an, So sehe ich, was sie getan; Betracht ich meine Siebensachen, Seh ich, was ich h¨att sollen machen. –Goethe, Weimar 1814/1815 1. Introduction. It is a virtually impossible task to do justice, in a short span of time and space, to the great genius of Leonhard Euler. All we can do, in this lecture, is to bring across some glimpses of Euler’s incredibly voluminous and diverse work, which today fills 74 massive volumes of the Opera omnia (with two more to come). Nine additional volumes of correspondence are planned and have already appeared in part, and about seven volumes of notebooks and diaries still await editing! We begin in section 2 with a brief outline of Euler’s life, going through the three stations of his life: Basel, St.
    [Show full text]
  • Hamilton's Principle in Continuum Mechanics
    Hamilton’s Principle in Continuum Mechanics A. Bedford University of Texas at Austin This document contains the complete text of the monograph published in 1985 by Pitman Publishing, Ltd. Copyright by A. Bedford. 1 Contents Preface 4 1 Mechanics of Systems of Particles 8 1.1 The First Problem of the Calculus of Variations . 8 1.2 Conservative Systems . 12 1.2.1 Hamilton’s principle . 12 1.2.2 Constraints.......................... 15 1.3 Nonconservative Systems . 17 2 Foundations of Continuum Mechanics 20 2.1 Mathematical Preliminaries . 20 2.1.1 Inner Product Spaces . 20 2.1.2 Linear Transformations . 22 2.1.3 Functions, Continuity, and Differentiability . 24 2.1.4 Fields and the Divergence Theorem . 25 2.2 Motion and Deformation . 27 2.3 The Comparison Motion . 32 2.4 The Fundamental Lemmas . 36 3 Mechanics of Continuous Media 39 3.1 The Classical Theories . 40 3.1.1 IdealFluids.......................... 40 3.1.2 ElasticSolids......................... 46 3.1.3 Inelastic Materials . 50 3.2 Theories with Microstructure . 54 3.2.1 Granular Solids . 54 3.2.2 Elastic Solids with Microstructure . 59 2 4 Mechanics of Mixtures 65 4.1 Motions and Comparison Motions of a Mixture . 66 4.1.1 Motions............................ 66 4.1.2 Comparison Fields . 68 4.2 Mixtures of Ideal Fluids . 71 4.2.1 Compressible Fluids . 71 4.2.2 Incompressible Fluids . 73 4.2.3 Fluids with Microinertia . 75 4.3 Mixture of an Ideal Fluid and an Elastic Solid . 83 4.4 A Theory of Mixtures with Microstructure . 86 5 Discontinuous Fields 91 5.1 Singular Surfaces .
    [Show full text]
  • Euler and Chebyshev: from the Sphere to the Plane and Backwards Athanase Papadopoulos
    Euler and Chebyshev: From the sphere to the plane and backwards Athanase Papadopoulos To cite this version: Athanase Papadopoulos. Euler and Chebyshev: From the sphere to the plane and backwards. 2016. hal-01352229 HAL Id: hal-01352229 https://hal.archives-ouvertes.fr/hal-01352229 Preprint submitted on 6 Aug 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. EULER AND CHEBYSHEV: FROM THE SPHERE TO THE PLANE AND BACKWARDS ATHANASE PAPADOPOULOS Abstract. We report on the works of Euler and Chebyshev on the drawing of geographical maps. We point out relations with questions about the fitting of garments that were studied by Chebyshev. This paper will appear in the Proceedings in Cybernetics, a volume dedicated to the 70th anniversary of Academician Vladimir Betelin. Keywords: Chebyshev, Euler, surfaces, conformal mappings, cartography, fitting of garments, linkages. AMS classification: 30C20, 91D20, 01A55, 01A50, 53-03, 53-02, 53A05, 53C42, 53A25. 1. Introduction Euler and Chebyshev were both interested in almost all problems in pure and applied mathematics and in engineering, including the conception of industrial ma- chines and technological devices. In this paper, we report on the problem of drawing geographical maps on which they both worked.
    [Show full text]
  • Leonhard Euler - Wikipedia, the Free Encyclopedia Page 1 of 14
    Leonhard Euler - Wikipedia, the free encyclopedia Page 1 of 14 Leonhard Euler From Wikipedia, the free encyclopedia Leonhard Euler ( German pronunciation: [l]; English Leonhard Euler approximation, "Oiler" [1] 15 April 1707 – 18 September 1783) was a pioneering Swiss mathematician and physicist. He made important discoveries in fields as diverse as infinitesimal calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[2] He is also renowned for his work in mechanics, fluid dynamics, optics, and astronomy. Euler spent most of his adult life in St. Petersburg, Russia, and in Berlin, Prussia. He is considered to be the preeminent mathematician of the 18th century, and one of the greatest of all time. He is also one of the most prolific mathematicians ever; his collected works fill 60–80 quarto volumes. [3] A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: "Read Euler, read Euler, he is our teacher in all things," which has also been translated as "Read Portrait by Emanuel Handmann 1756(?) Euler, read Euler, he is the master of us all." [4] Born 15 April 1707 Euler was featured on the sixth series of the Swiss 10- Basel, Switzerland franc banknote and on numerous Swiss, German, and Died Russian postage stamps. The asteroid 2002 Euler was 18 September 1783 (aged 76) named in his honor. He is also commemorated by the [OS: 7 September 1783] Lutheran Church on their Calendar of Saints on 24 St. Petersburg, Russia May – he was a devout Christian (and believer in Residence Prussia, Russia biblical inerrancy) who wrote apologetics and argued Switzerland [5] forcefully against the prominent atheists of his time.
    [Show full text]
  • Coefficient of Restitution Interpreted As Damping in Vibroimpact K
    Coefficient of restitution interpreted as damping in vibroimpact Kenneth Hunt, Erskine Crossley To cite this version: Kenneth Hunt, Erskine Crossley. Coefficient of restitution interpreted as damping in vibroimpact. Journal of Applied Mechanics, American Society of Mechanical Engineers, 1975, 10.1115/1.3423596. hal-01333795 HAL Id: hal-01333795 https://hal.archives-ouvertes.fr/hal-01333795 Submitted on 19 Jun 2016 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Coefficient of Restitution Interpreted as Damping in Vibroimpact K. H. Hunt During impact the relative motion of two bodies is often taken to be simply represented Dean and Professor of Engineering, as half of a damped sine wave, according to the Kelvin-Voigt model. This is shown to be Monash University, logically untenable, for it indicates that the bodies must exert tension on one another Clayton, VIctoria, Australia just before separating. Furthermore, it denotes that the damping energy loss is propor­ tional to the square of the impactin{f velocity, instead of to its cube, as can be deduced F. R. E. Crossley from Goldsmith's work. A damping term h"i is here introduced; for a sphere impacting Professor, a plate Hertz gives n = 3/2.
    [Show full text]
  • Siméon-Denis Poisson Mathematics in the Service of Science
    S IMÉ ON-D E N I S P OISSON M ATHEMATICS I N T H E S ERVICE O F S CIENCE E XHIBITION AT THE MATHEMATICS LIBRARY U NIVE RSIT Y O F I L L I N O I S A T U RBANA - C HAMPAIGN A U G U S T 2014 Exhibition on display in the Mathematics Library of the University of Illinois at Urbana-Champaign 4 August to 14 August 2014 in association with the Poisson 2014 Conference and based on SIMEON-DENIS POISSON, LES MATHEMATIQUES AU SERVICE DE LA SCIENCE an exhibition at the Mathematics and Computer Science Research Library at the Université Pierre et Marie Curie in Paris (MIR at UPMC) 19 March to 19 June 2014 Cover Illustration: Portrait of Siméon-Denis Poisson by E. Marcellot, 1804 © Collections École Polytechnique Revised edition, February 2015 Siméon-Denis Poisson. Mathematics in the Service of Science—Exhibition at the Mathematics Library UIUC (2014) SIMÉON-DENIS POISSON (1781-1840) It is not too difficult to remember the important dates in Siméon-Denis Poisson’s life. He was seventeen in 1798 when he placed first on the entrance examination for the École Polytechnique, which the Revolution had created four years earlier. His subsequent career as a “teacher-scholar” spanned the years 1800-1840. His first publications appeared in the Journal de l’École Polytechnique in 1801, and he died in 1840. Assistant Professor at the École Polytechnique in 1802, he was named Professor in 1806, and then, in 1809, became a professor at the newly created Faculty of Sciences of the Université de Paris.
    [Show full text]
  • Applied Mechanics - 3300008 Unit 1: Introduction 1
    MECHANICAL ENGG. DEPT SEMESTER #2 APPLIED MECHANICS - 3300008 UNIT 1: INTRODUCTION 1. Differentiate: 1) Vector quantity and Scalar quantity 2) Kinetics and Kinematics 2. State SI system unit of following quantities: 1) Density 2) Angle 3) Work 4) Power 5) Force 6) Pressure 7) Velocity 8) Torque 3. Name the type of quantities: 1) Density 2) Time 3) Work 4) Energy 5) Force 6) Mass 7) Velocity 8) Speed UNIT 2: COPLANAR CONCURRENT FORCES 1. Explain principle of super position of forces. 2. Explain principle of transmissibility of forces. 3. State and prove lami’s theorem. State its limitation. 4. Explain polygon law of forces. 5. Classify forces. 6. State and Explain law of parallelogram of forces. 7. State and Explain law of triangle of forces. 8. The system shown in figure-1 is in equilibrium. Find unknown forces P and Q. 9. Find magnitude and direction of resultant force for fig.2 Fig.1 Fig. 2 Fig.3 Fig.4 10. A load of 75 KN is hung by means of a rope attached to a hook in horizontal ceiling. What horizontal force should be applied so that rope makes 60o with the ceiling? Page 1 of 27 MECHANICAL ENGG. DEPT SEMESTER #2 11. The boy in a garden holding two chains in his hand which are Hooked with Horizontal steel bar making an angel 65° & other chain with an angle of 55° With the steel bar, if the weight of boy is 55 kg, than find tension developed in both chain. 12. A circular sphere weighing 500 N and having a radius of 200mm hangs by a string AC 400 mm long as shown in Figure 3.
    [Show full text]
  • Analytical Continuum Mechanics À La Hamilton-Piola: Least Action Principle for Second Gradient Continua and Capillary Fluids Nicolas Auffray, F
    Analytical continuum mechanics à la Hamilton-Piola: least action principle for second gradient continua and capillary fluids Nicolas Auffray, F. Dell’Isola, V. Eremeyev, A. Madeo, G.Rosi To cite this version: Nicolas Auffray, F. Dell’Isola, V. Eremeyev, A. Madeo, G. Rosi. Analytical continuum mechanics àla Hamilton-Piola: least action principle for second gradient continua and capillary fluids. Mathematics and Mechanics of Solids, SAGE Publications, 2013. hal-00836085 HAL Id: hal-00836085 https://hal.archives-ouvertes.fr/hal-00836085 Submitted on 21 Jun 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Analytical continuum mechanics à la Hamilton-Piola: least action principle for second gradient continua and capillary fluids By N. Auffraya, F. dell’Isolab, V. Eremeyevc, A. Madeod and G. Rosie a Université Paris-Est, Laboratoire Modélisation et Simulation Multi Echelle, MSME UMR 8208 CNRS, 5 bd Descartes, 77454 Marne-la-Vallée, France bDipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma La Sapienza, Via Eudossiana 18, 00184, Roma,
    [Show full text]
  • Lagrangian Mechanics - Wikipedia, the Free Encyclopedia Page 1 of 11
    Lagrangian mechanics - Wikipedia, the free encyclopedia Page 1 of 11 Lagrangian mechanics From Wikipedia, the free encyclopedia Lagrangian mechanics is a re-formulation of classical mechanics that combines Classical mechanics conservation of momentum with conservation of energy. It was introduced by the French mathematician Joseph-Louis Lagrange in 1788. Newton's Second Law In Lagrangian mechanics, the trajectory of a system of particles is derived by solving History of classical mechanics · the Lagrange equations in one of two forms, either the Lagrange equations of the Timeline of classical mechanics [1] first kind , which treat constraints explicitly as extra equations, often using Branches [2][3] Lagrange multipliers; or the Lagrange equations of the second kind , which Statics · Dynamics / Kinetics · Kinematics · [1] incorporate the constraints directly by judicious choice of generalized coordinates. Applied mechanics · Celestial mechanics · [4] The fundamental lemma of the calculus of variations shows that solving the Continuum mechanics · Lagrange equations is equivalent to finding the path for which the action functional is Statistical mechanics stationary, a quantity that is the integral of the Lagrangian over time. Formulations The use of generalized coordinates may considerably simplify a system's analysis. Newtonian mechanics (Vectorial For example, consider a small frictionless bead traveling in a groove. If one is tracking the bead as a particle, calculation of the motion of the bead using Newtonian mechanics) mechanics would require solving for the time-varying constraint force required to Analytical mechanics: keep the bead in the groove. For the same problem using Lagrangian mechanics, one Lagrangian mechanics looks at the path of the groove and chooses a set of independent generalized Hamiltonian mechanics coordinates that completely characterize the possible motion of the bead.
    [Show full text]
  • Least Action Principle Applied to a Non-Linear Damped Pendulum Katherine Rhodes Montclair State University
    Montclair State University Montclair State University Digital Commons Theses, Dissertations and Culminating Projects 1-2019 Least Action Principle Applied to a Non-Linear Damped Pendulum Katherine Rhodes Montclair State University Follow this and additional works at: https://digitalcommons.montclair.edu/etd Part of the Applied Mathematics Commons Recommended Citation Rhodes, Katherine, "Least Action Principle Applied to a Non-Linear Damped Pendulum" (2019). Theses, Dissertations and Culminating Projects. 229. https://digitalcommons.montclair.edu/etd/229 This Thesis is brought to you for free and open access by Montclair State University Digital Commons. It has been accepted for inclusion in Theses, Dissertations and Culminating Projects by an authorized administrator of Montclair State University Digital Commons. For more information, please contact [email protected]. Abstract The principle of least action is a variational principle that states an object will always take the path of least action as compared to any other conceivable path. This principle can be used to derive the equations of motion of many systems, and therefore provides a unifying equation that has been applied in many fields of physics and mathematics. Hamilton’s formulation of the principle of least action typically only accounts for conservative forces, but can be reformulated to include non-conservative forces such as friction. However, it can be shown that with large values of damping, the object will no longer take the path of least action. Through numerical
    [Show full text]
  • Certain Interesting Properties of Action and Its Application Towards Achieving Greater Organization in Complex Systems
    Certain Interesting Properties of Action and Its Application Towards Achieving Greater Organization in Complex Systems Atanu Bikash Chatterjee Department of Mechanical Engineering, Bhilai Institute of Technology, Bhilai House Durg, 490006 Chhattisgarh, India e -mail: [email protected] Abstract: The Principle of Least Action has evolved and established itself as the most basic law of physics. This allows us to see how this fundamental law of nature determines the development of the system towards states with less action, i.e., organized states. A system undergoing a natural process is formulated as a game that tends to organize the system in the least possible time. Also, other concepts of game theory are related to their profound physical counterparts. Although no fundamentally new findings are provided, it is quite interesting to see certain important properties of a complex system and their far-reaching consequences. Keywords: Action, dead state, extensive property, super-structure, system, organization, complexity, Nash-equilibrium strategy, pay-off function. Introduction: All processes occurring in the universe are rooted in physics and have a physical explanation. All the structures in the universe exist, because they are in their state of least action or tend towards it. All natural process are directed along the steepest descents of an energy landscape by equalizing differences in energy via various transport and transformation processes, e.g., diffusion, heat flows, electric currents and chemical reactions [1]. In any system, simple or complex, the system spontaneously calculates which path will use least effort for that process. In an organized system, the action of a single element will not be at minimum, because of the constraints, but the sum of all actions in the whole system will be at minimum [2].
    [Show full text]
  • MEC 119.3 Applied Mechanics (3-2-0)
    MEC 119.3 Applied Mechanics (3-2-0) Theory Practical Total Sessional 50 - 50 Final 50 - 50 Total 100 - 100 Course Objectives: 1. To develop knowledge of mechanical equilibrium 2. To understand Newton’s Laws of motion. 3. To apply mechanics in wide range of engineering applications. Course Contents: 1. Introduction ( 2 hrs) Engineering Mechanics: definition and scope, Concept of statics and dynamics, Concept of particle and rigid bodies, System of units. 2. Forces on Particles and Rigid Bodies (5 hrs) Characteristics of Forces, Types of Forces, Transmissibility and equivalent Forces, Resolution and Composition of Forces, Moment of a Forces, Resolution and Composition of Forces, Moment of a Force about a point, Moment of a Force About an Axis, Theory of Couples, Resolution of a Force into a Force and a Couple, Resultant of a System of Forces. 3. Equations of Equations of Equilibrium (3 hrs) The free body diagram, General equations of equilibrium, equations of equilibrium for various forces systems. 4. Distributed Forces (3 hrs) Center of Gravity an d Centroid of area, line and volume. Second moment of area, parallel axis theorem, polar moment of inertia, radius of gyration. 5. Friction Forces (2 hrs) Laws of Friction, Coefficients of Friction and angled of friction. 6. Kinematics of Particles (6 hrs) Rectilinear motion of particles, Curvilinear motion of particle, Rectangular components of velocity and acceleration, Tangential and normal components, Radial and transverse components, Simple relative motion. 7. Kinetics of Particles (4 hrs) Newton’s second law of motion, linear momentum of a particle, Equations of motion, Dynamic equilibrium, Angular momentum of a particle, Motion due to central force and conservation of angular momentum.
    [Show full text]