deformationcontinuation

Deformation continuation is a numerical method used in computational mathematics and engineering to solve nonlinear equations and systems of equations. It is particularly useful when traditional methods, such as Newton-Raphson, fail due to the presence of multiple solutions or singularities. The method involves gradually deforming a simple problem, for which a solution is known, into the original problem of interest. This deformation is typically parameterized by a continuation parameter, which is gradually varied from an initial value to a final value.

The basic idea behind deformation continuation is to construct a family of problems that interpolates between the simple problem and the original problem. By solving this family of problems sequentially, one can track the solution path from the known solution of the simple problem to the solution of the original problem. This approach allows for the detection and resolution of bifurcations and singularities that might otherwise cause traditional methods to fail.

Deformation continuation can be applied to a wide range of problems, including nonlinear optimization, bifurcation analysis, and the solution of nonlinear partial differential equations. It is particularly useful in engineering applications, such as structural analysis, fluid dynamics, and control systems, where nonlinearities are common. The method can be implemented using various numerical techniques, such as pseudo-arclength continuation and homotopy methods, depending on the specific problem and the desired accuracy.

One of the key advantages of deformation continuation is its ability to provide a global view of the solution space, rather than just a local solution. This can be particularly valuable in problems with multiple solutions or complex solution structures. However, the method can be computationally intensive, especially for high-dimensional problems, and may require careful selection of the continuation parameter and the deformation path. Despite these challenges, deformation continuation remains a powerful tool for solving nonlinear equations and systems of equations in a wide range of applications.