A mesh refinement method is described for solving a continuous-time optimal control problem using... more A mesh refinement method is described for solving a continuous-time optimal control problem using collocation at Legendre-Gauss-Radau points. The method allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. First, a relative error estimate is derived based on the difference between the Lagrange polynomial approximation of the state and a Legendre-Gauss-Radau quadrature integration of the dynamics within a mesh interval. The derived relative error estimate is then used to decide if the degree of the approximating polynomial within a mesh should be increased or if the mesh interval should be divided into sub-intervals. The degree of the approximating polynomial within a mesh interval is increased if the polynomial degree estimated by the method remains below a maximum allowable degree. Otherwise, the mesh interval is divided into sub-intervals. The process of refining the mesh is repeated until a specified relative error tolerance is met. Three examples highlight various features of the method and show that the approach is more computationally efficient and produces significantly smaller mesh sizes for a given accuracy tolerance when compared with fixed-order methods.
Finite element analysis (FEA) is a computer based numerical method for solving problems in a wide... more Finite element analysis (FEA) is a computer based numerical method for solving problems in a wide range of engineering areas such as stress analysis, thermal analysis and fluid flow, diffusion, and magnetic field interactions.
UNIT I FRICTION 7
Topography of Surfaces – Surface features – Properties and measurement – Surfac... more UNIT I FRICTION 7 Topography of Surfaces – Surface features – Properties and measurement – Surface interaction – Adhesive Theory of Sliding Friction – Rolling Friction – Friction properties of metallic and non metallic materials – Friction in extreme conditions – Thermal considerations in sliding contact UNIT II WEAR 6 Introduction – Abrasive wear, Erosive, Cavitation, Adhesion, Fatigue wear and Fretting Wear- Laws of wear – Theoretical wear models – Wear of metals and non metals – International standards in friction and wear measurements UNIT III CORROSION 10 Introduction – Principle of corrosion – Classification of corrosion – Types of corrosion – Factors influencing corrosion – Testing of corrosion – In-service monitoring, Simulated service, Laboratory testing – Evaluation of corrosion – Prevention of Corrosion – Material sele
The governing equations for problems solved by the finite element method are typically formulated... more The governing equations for problems solved by the finite element method are typically formulated by partial differential equations in their original form. These are rewritten into a weak form, such that domain integration can be utilized to satisfy the governing equations in an average sense. A functional is set up for the system, typically describing the energy or energy rate and implying that the solution can be found by minimization. For a generic functional, this is written as where the functional is a function of the coordinates x i and the primary variable u i being e.g. displacements or velocities for mechanical problems depending on the formulation. The domain integration is approximated by a summation over a finite number of elements discretizing the domain. Figure 2.1 illustrates a three-dimensional domain discretized by hexahedral elements with eight nodes. The variables are defined and solved in the nodal points, and evaluation of variables in the domain is performed by interpolation in each element. Shared nodes give rise to an assembly of elements into a global system of equations of the form where K is the stiffness matrix, u is the primary variable and f is the applied load, e.g. stemming from applied tractions F on a surface S F in Fig. 2.1. The system of equations (2.2) is furthermore subject to essential boundary conditions, e.g. prescribed displacements or velocities u along a surface S U. The basic aspects of available finite element formulations in terms of mod-eling and computation are briefly reviewed in this chapter. This will support the choice of formulation to be detailed and applied in the remaining chapters, where an electro-thermo-mechanical finite element formulation is presented (2.1) ∂� ∂u = ∂ ∂u � V f(x i , u i) dV = ∂ ∂u � j f(x i , u i) ∆V j = 0 (2.2) Ku = f Finite Element Formulations Chapter 2
A mesh refinement method is described for solving a continuous-time optimal control problem using... more A mesh refinement method is described for solving a continuous-time optimal control problem using collocation at Legendre-Gauss-Radau points. The method allows for changes in both the number of mesh intervals and the degree of the approximating polynomial within a mesh interval. First, a relative error estimate is derived based on the difference between the Lagrange polynomial approximation of the state and a Legendre-Gauss-Radau quadrature integration of the dynamics within a mesh interval. The derived relative error estimate is then used to decide if the degree of the approximating polynomial within a mesh should be increased or if the mesh interval should be divided into sub-intervals. The degree of the approximating polynomial within a mesh interval is increased if the polynomial degree estimated by the method remains below a maximum allowable degree. Otherwise, the mesh interval is divided into sub-intervals. The process of refining the mesh is repeated until a specified relative error tolerance is met. Three examples highlight various features of the method and show that the approach is more computationally efficient and produces significantly smaller mesh sizes for a given accuracy tolerance when compared with fixed-order methods.
Finite element analysis (FEA) is a computer based numerical method for solving problems in a wide... more Finite element analysis (FEA) is a computer based numerical method for solving problems in a wide range of engineering areas such as stress analysis, thermal analysis and fluid flow, diffusion, and magnetic field interactions.
UNIT I FRICTION 7
Topography of Surfaces – Surface features – Properties and measurement – Surfac... more UNIT I FRICTION 7 Topography of Surfaces – Surface features – Properties and measurement – Surface interaction – Adhesive Theory of Sliding Friction – Rolling Friction – Friction properties of metallic and non metallic materials – Friction in extreme conditions – Thermal considerations in sliding contact UNIT II WEAR 6 Introduction – Abrasive wear, Erosive, Cavitation, Adhesion, Fatigue wear and Fretting Wear- Laws of wear – Theoretical wear models – Wear of metals and non metals – International standards in friction and wear measurements UNIT III CORROSION 10 Introduction – Principle of corrosion – Classification of corrosion – Types of corrosion – Factors influencing corrosion – Testing of corrosion – In-service monitoring, Simulated service, Laboratory testing – Evaluation of corrosion – Prevention of Corrosion – Material sele
The governing equations for problems solved by the finite element method are typically formulated... more The governing equations for problems solved by the finite element method are typically formulated by partial differential equations in their original form. These are rewritten into a weak form, such that domain integration can be utilized to satisfy the governing equations in an average sense. A functional is set up for the system, typically describing the energy or energy rate and implying that the solution can be found by minimization. For a generic functional, this is written as where the functional is a function of the coordinates x i and the primary variable u i being e.g. displacements or velocities for mechanical problems depending on the formulation. The domain integration is approximated by a summation over a finite number of elements discretizing the domain. Figure 2.1 illustrates a three-dimensional domain discretized by hexahedral elements with eight nodes. The variables are defined and solved in the nodal points, and evaluation of variables in the domain is performed by interpolation in each element. Shared nodes give rise to an assembly of elements into a global system of equations of the form where K is the stiffness matrix, u is the primary variable and f is the applied load, e.g. stemming from applied tractions F on a surface S F in Fig. 2.1. The system of equations (2.2) is furthermore subject to essential boundary conditions, e.g. prescribed displacements or velocities u along a surface S U. The basic aspects of available finite element formulations in terms of mod-eling and computation are briefly reviewed in this chapter. This will support the choice of formulation to be detailed and applied in the remaining chapters, where an electro-thermo-mechanical finite element formulation is presented (2.1) ∂� ∂u = ∂ ∂u � V f(x i , u i) dV = ∂ ∂u � j f(x i , u i) ∆V j = 0 (2.2) Ku = f Finite Element Formulations Chapter 2
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Papers by Sedhu Pathi
Topography of Surfaces – Surface features – Properties and measurement – Surface interaction – Adhesive Theory of Sliding Friction – Rolling Friction – Friction properties of metallic and non metallic materials – Friction in extreme conditions – Thermal considerations in sliding contact
UNIT II WEAR 6
Introduction – Abrasive wear, Erosive, Cavitation, Adhesion, Fatigue wear and Fretting Wear- Laws of wear – Theoretical wear models – Wear of metals and non metals – International standards in friction and wear measurements
UNIT III CORROSION 10
Introduction – Principle of corrosion – Classification of corrosion – Types of corrosion – Factors influencing corrosion – Testing of corrosion – In-service monitoring, Simulated service, Laboratory testing – Evaluation of corrosion – Prevention of Corrosion – Material sele
Topography of Surfaces – Surface features – Properties and measurement – Surface interaction – Adhesive Theory of Sliding Friction – Rolling Friction – Friction properties of metallic and non metallic materials – Friction in extreme conditions – Thermal considerations in sliding contact
UNIT II WEAR 6
Introduction – Abrasive wear, Erosive, Cavitation, Adhesion, Fatigue wear and Fretting Wear- Laws of wear – Theoretical wear models – Wear of metals and non metals – International standards in friction and wear measurements
UNIT III CORROSION 10
Introduction – Principle of corrosion – Classification of corrosion – Types of corrosion – Factors influencing corrosion – Testing of corrosion – In-service monitoring, Simulated service, Laboratory testing – Evaluation of corrosion – Prevention of Corrosion – Material sele