Solving viscoelastic problems with a Laplace transform approach supplanted by ARX models, suggesting a way to upgrade Finite Element or spectral codes
Finite Element codes used for solving the mechanical equilibrium equations in transient problems associated to (time-dependent) viscoelastic media generally relies on time-discretized versions of the selected constitutive law. Recent concerns about the use of non-integer differential equations to describe viscoelasticity or well-founded ideas based upon the use of a behavior's law directly derived from Dynamic Mechanical Analysis (DMA) experiments in frequency domain, could make the Laplace domain approach particularly attractive if embedded in a time discretized scheme. Based upon the inversion of Laplace transforms, this paper shows that this aim is not only possible but also gives rise to a simple algorithm having good performances in terms of computation times and precision. Such an approach, which fully relies on the Laplace-defined Behavioral Transfer Function (LTBF) can be promoted if it uses AutoRegressive with eXogeneous input parametric models perfectly substitutable to the real LTBF. They avoid the hitherto prohibitive pitfall of having to store all past data in the computer's memory while maintaining an equal computation precision.
10 octobre 2023
Efficient Computation of Redundancy Matrices for Moderately Redundant Truss and Frame Structures
Large statically indeterminate truss and frame structures exhibit complex load-bearing behavior, and redundancy matrices are helpful for their analysis and design. Depending on the task, the full redundancy matrix or only its diagonal entries are required. The standard computation procedure has a high computational effort. Many structures fall in the category of moderately redundant, i.e., the ratio of the statical indeterminacy to the number of all load-carrying modes of all elements is less one half. This paper proposes a closed-form expression for redundancy contributions that is computationally efficient for moderately redundant systems. The expression is derived via a factorization of the redundancy matrix that is based on singular value decomposition. Several examples illustrate the behavior of the method for increasing size of systems and, where applicable, for increasing degree of statical indeterminacy.
27 septembre 2023