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  2. Penalty method - Wikipedia

    en.wikipedia.org/wiki/Penalty_method

    In the above equations, (()) is the exterior penalty function while is the penalty coefficient. When the penalty coefficient is 0, f p = f . In each iteration of the method, we increase the penalty coefficient p {\displaystyle p} (e.g. by a factor of 10), solve the unconstrained problem and use the solution as the initial guess for the next ...

  3. Meshfree methods - Wikipedia

    en.wikipedia.org/wiki/Meshfree_methods

    Several methods have been developed to impose the essential boundary conditions weakly, including Lagrange multipliers, Nitche's method, and the penalty method. As for quadrature , nodal integration is generally preferred which offers simplicity, efficiency, and keeps the meshfree method free of any mesh (as opposed to using Gauss quadrature ...

  4. Perturbation theory - Wikipedia

    en.wikipedia.org/wiki/Perturbation_theory

    Perturbation theory has been used in a large number of different settings in physics and applied mathematics. Examples of the "collection of equations" include algebraic equations, [6] differential equations [7] (e.g., the equations of motion [8] and commonly wave equations), thermodynamic free energy in statistical mechanics, radiative ...

  5. Constrained optimization - Wikipedia

    en.wikipedia.org/wiki/Constrained_optimization

    Many constrained optimization algorithms can be adapted to the unconstrained case, often via the use of a penalty method. However, search steps taken by the unconstrained method may be unacceptable for the constrained problem, leading to a lack of convergence. This is referred to as the Maratos effect. [3]

  6. Lagrangian relaxation - Wikipedia

    en.wikipedia.org/wiki/Lagrangian_relaxation

    The penalty method does not use dual variables but rather removes the constraints and instead penalizes deviations from the constraint. The method is conceptually simple but usually augmented Lagrangian methods are preferred in practice since the penalty method suffers from ill-conditioning issues.

  7. Perturbation theory (quantum mechanics) - Wikipedia

    en.wikipedia.org/wiki/Perturbation_theory...

    These corrections, being small compared to the size of the quantities themselves, can be calculated using approximate methods such as asymptotic series. The complicated system can therefore be studied based on knowledge of the simpler one. In effect, it is describing a complicated unsolved system using a simple, solvable system.

  8. Lagrangian mechanics - Wikipedia

    en.wikipedia.org/wiki/Lagrangian_mechanics

    For example, in calculation of the motion of a torus rolling on a horizontal surface with a pearl sliding inside, the time-varying constraint forces like the angular velocity of the torus, motion of the pearl in relation to the torus made it difficult to determine the motion of the torus with Newton's equations. [7]

  9. Coefficient of restitution - Wikipedia

    en.wikipedia.org/wiki/Coefficient_of_restitution

    The COR is a property of a pair of objects in a collision, not a single object. If a given object collides with two different objects, each collision has its own COR. When a single object is described as having a given coefficient of restitution, as if it were an intrinsic property without reference to a second object, some assumptions have been made – for example that the collision is with ...