From Stress to Shape: Equilibrium of Cloister and Cross Vaults

The assessment of the equilibrium and the safety of masonry vaults is of high relevance for the conservation and restoration of historical heritage. In the literature many approaches have been proposed for this tasks, starting from the 17th century. In this work we focus on the Membrane Equilibrium Analysis, developed under the Heyman’s theory of Limit Analysis. Within this theory, the equilibrium of a vault is assessed if it is possible to find at least one membrane surface, between the volume of the vaults, being in equilibrium under the given loads through a purely compressive stress field. The equilibrium of membranes is described by a second order partial differential equation, which is definitely elliptic only when a negative semidefinite stress is assigned, and the shape is the unknown of the problem. The proposed algorithm aims at finding membrane shapes, entirely comprised between the geometry of the vault, in equilibrium with admissible stress fields, through the minimization of an error function with respect to shape parameters of the stress potential, and then, with respect to the boundary values of the membrane shape. The application to two test cases shows the viability of this tool for the assessment of the equilibrium of existing masonry vaults.

Principal Pivot Transforms of Quasidefinite Matrices and Semidefinite Lagrangian Subspaces

Lagrangian subspaces are linear subspaces that appear naturally in control theory applications, and especially in the context of algebraic Riccati equations. We introduce a class of semidefinite Lagrangian subspaces and show that these subspaces can be represented by a subset I ⊆ {1, 2, . . . , n} and a Hermitian matrix X ∈ C n×n with the property that the submatrix X II is negative semidefinite and the submatrix X I c I c is positive semidefinite. A matrix X with these definiteness properties is called I-semidefinite and it is a generalization of a quasidefinite matrix. Under mild hypotheses which hold true in most applications, the Lagrangian subspace associated to the stabilizing solution of an algebraic Riccati equation is semidefinite, and in addition we show that there is a bijection between Hamiltonian and symplectic pencils and semidefinite Lagrangian subspaces; hence this structure is ubiquitous in control theory. The (symmetric) principal pivot transform (PPT) is a map used by Mehrmann and Poloni [SIAM J. Matrix Anal. Appl., 33(2012), pp. 780–805] to convert between two different pairs (I, X) and (J , X 0 ) representing the same Lagrangian subspace. For a semidefinite Lagrangian subspace, we prove that the symmetric PPT of an I-semidefinite matrix X is a J -semidefinite matrix X 0 , and we derive an implementation of the transformation X 7→ X 0 that both makes use of the definiteness properties of X and guarantees the definiteness of the submatrices of X 0 in finite arithmetic. We use the resulting formulas to obtain a semidefiniteness-preserving version of an optimization algorithm introduced by Mehrmann and Poloni to compute a pair (I opt , X opt ) with M = max i,j |(X opt ) ij | as small as possible. Using semidefiniteness allows one to obtain a stronger inequality on M with respect to the general case.

On the Global Stability of a Class of Nonlinear Time-Varying Systems

In this paper the problem of the stability of motion of the equilibrium solution x1 = x2… = xn = 0 is studied, in the sense of Lyapunov, for a class of systems represented by a system of differential equations dxi/dt = Fi (x1, x2…xn, t), i = 1, 2…n or x˙ = A (x,t)x. Various x1 are known as state variables and Fi (0, 0…0, ∞) = 0. The various elements of square matrix A (x, t) are functions of time as well as functions of state variables x. Two different methods for generating Lyapunov functions are developed. In the first method the differential equation is multiplied by various state variables and integrated by parts to generate a proper Lyapunov function and a number of matrices α, α1…αn, S1, S2…Sn. The second method assumes a quadratic Lyapunov function V = [x′S(x,t)x], x′ being the transpose of x. The elements of S(x,t) may be functions of time and the state variables or constants. The time derivative V˙ is given by V˙ = x′[B′A + S˙]x = x′T(t,x)x where B x gives the gradient ∇V, and S˙ is defined as ∂S/∂t. For the equilibrium solution x1 = x2… = xn = 0 to be stable it is required that V˙ should be negative definite or negative semidefinite and V should be positive definite. These considerations determine the sufficient conditions of stability.