![]() ![]() In the table above, external constraints to the ground (i.e. Table 1: External constraints to the ground These constraints are usually depicted as follows: Symbol For instance, frame structures are based on the beam theory, and the related finite element has 6 degrees of freedom (3 displacements 3 rotations in a 3D space).įor a 2D problem, each node of the boundary has 3 degrees of freedom on which Dirichlet boundary conditions can be applied: 2 displacements (\(u_x\) and \(u_y\)) and 1 rotation (\(\omega\)). Structural mechanics is often based on formulations which include relative rotations, whose numerical resolution requires nonlinear shape functions in the finite element approximation. Thus, Dirichlet boundary conditions usually consist of imposing the displacement of the structure at given points. Solid mechanics is usually modeled through a displacement-based model. Applications of Boundary Conditions Structural and Solid Mechanics 1 Neumann B.C.), while the Robin condition implies only one constraint on the linear combination of the unknown function and its derivatives. It differs from the Robin condition because the Cauchy condition implies the imposition of two constraints (1 Dirichlet B.C. The Cauchy boundary condition is a condition on both the unknown field and its derivatives\(^5\). ![]() applied to different parts of the boundary. The mixed boundary condition differs from the Robin condition because the latter consists of different types of boundary conditions applied to the same region of the boundary, while the mixed condition implies different types of B.C. It is important to notice that boundary conditions must be applied on the whole boundary: the “free” boundary is anyways subjected to a homogeneous Neumann condition. The mixed boundary condition consists of applying different types of boundary conditions in different parts of the domain. This condition is also called the “ impedance condition“. Where \(a\) and \(b\) are real parameters. Given, for example, the Laplace equation, the boundary value problem with the Dirichlet b.c. This condition specifies the value that the unknown function needs to take on along the boundary of the domain. Robin boundary condition (also known as Type III).Neumann boundary condition (also known as Type II).Dirichlet boundary condition (also known as Type I).There are five types of boundary conditions: may lead to the divergence of the solution or to the convergence of a wrong solution. The choice of the boundary condition is fundamental for the resolution of the computational problem: a bad imposition of B.C. Different types of boundary conditions can be imposed on the boundary of the domain (Figure 1). Types of Boundary Conditionsīoth ordinary and partial differential equations require solving boundary conditions (B.C.). The Sturm-Liouville theory is extremely important for any computational problem because it enables us to understand if a problem is “well-posed” and how it is possible to obtain the solution. They studied the conditions that guarantee the existence and uniqueness of the solution of the differential problem and how it is affected by the boundary conditions\(^2\). They arise naturally in every problem based on a differential equation to be solved in space, while initial value problems usually refer to problems to be solved in time.īoundary value problems have been extensively studied by Jacques Charles François Sturm (1803-1855) and Joseph Liouville (1809-1882), who studied the eigenvalues of a linear differential equation of the second order\(^1\). It is opposed to the “initial value problem”, in which only the conditions on one extreme of the interval are known.īoundary value problems are extremely important as they model a vast amount of phenomena and applications, from solid mechanics to heat transfer, fluid mechanics, and acoustic diffusion. A boundary value problem is a differential equation (or system of differential equations) to be solved in a domain on whose boundary a set of conditions is known. Boundary conditions are constraints necessary for the solution of a boundary value problem. ![]()
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