![]() ![]() ![]() But the weak form doesn’t necessarily hold true for every point in the domain, and the solution of a weak form needs only lesser mathematical requirements as compared to that of the strong form. A weak form is basically another interpretation of the GDE and boundary conditions. To adjust for the lack of a strong form, we develop what is known as the ‘weak form’ of the system. The Finite Element Methods, Finite Difference Methods etc… are examples of such approximate numerical methods. In such cases, we resort to numerical methods which gives an ‘approximate solution’ of the system. If the system under consideration has inherent complexities in the form of geometrical arbitrariness, loading pattern or boundary conditions, a strong form may not be ensured and hence getting an exact solution is impossible. As an example, the strong form of a 1-D bar element is given below.īut deriving a strong form is not realistically possible all the time. This means that the underlying statements within is applicable to every single point inside the system and solving the ‘strong form’ gives you an exact solution of the system. The combination of GDE and the Boundary (and/or Initial) Conditions constitutes the ‘ strong form‘ of the system. To solve such a GDE, we may need to have some extra details about the system, which are known as the boundary conditions or initial conditions of the system. The differential equation thus developed for a physical system is called the Governing Differential Equation (GDE). ![]() This is true for simple things such as heat transfer, fluid flow, population growth, compound interest to complex things such as pandemic spread, nuclear reaction etc… As a Civil/Mechanical Engineer, we are concerned about the problems coming under the domain of continuum mechanics, such as spring systems, linear elastic bodies etc… The real-world problem is converted into a suitable mathematical differential equation statement, solved to arrive at a mathematical solution, which is then physically interpreted so as to arrive at a real-world solution. Most physical processes happening in our environment can be expressed in terms of (partial) differential equations. ![]() Įuler and Lagrange also studied problems on a conditional extremum.Variational Calculus: Deriving the strong and weak form Is a vector function of arbitrary dimension. The principal results concerning the simplest problem of variational calculus are applied to the general case of functionals of the type Implies supplementary conditions to be satisfied by the mobile ends - the so-called transversality condition which, in conjunction with the boundary conditions, yields a closed system of conditions for the boundary value problem. In problems with mobile ends the condition $ \delta J = 0 $ It is required to minimize the functional The following scheme describes a rather wide range of problems of classical variational calculus. by the method of small perturbations of the arguments and functionals such problems, in the wider sense, are opposite to discrete optimization problems. The term "variational calculus" has a broader sense also, viz., a branch of the theory of extremal problems in which the extrema are studied by the "method of variations" (cf. This is the framework of the problems which are still known as problems of classical variational calculus. The branch of mathematics in which one studies methods for obtaining extrema of functionals which depend on the choice of one or several functions subject to constraints of various kinds (phase, differential, integral, etc.) imposed on these functions. ![]()
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