An offline-online strategy for multiscale problems with random defects

In this paper, we propose an offline-online strategy based on the Localized Orthogonal Decomposition (LOD) method for elliptic multiscale problems with randomly perturbed diffusion coefficient. We consider a periodic deterministic coefficient with local defects that occur with probability $p$. The offline phase pre-computes entries to global LOD stiffness matrices on a single reference element (exploiting the periodicity) for a selection of defect configurations. Given a sample of the perturbed diffusion the corresponding LOD stiffness matrix is then computed by taking linear combinations of the pre-computed entries, in the online phase. Our computable error estimates show that this yields a good approximation of the solution for small $p$, which is illustrated by extensive numerical experiments. This makes the proposed technique attractive already for moderate sample sizes in a Monte Carlo simulation.

Gramian-constrained optimization process for the stiffness model identification of industrial manipulators

This work deals with the elastostatic identification of industrial manipulators. By reviewing the basics of the physical elastic properties of both links and joints in the framework of the lumped stiffness modeling techniques, the Gramian nature of the stiffness matrices has been found out adequate to do so. Then, a novel optimization method has been developed, which incorporates the Gramian matrix formulation along a non-linear optimization process, acting as an intrinsic constraint for the conservativeness of the elastostatic modeling. Numerical and experimental analyses evince the effectiveness of the proposed method, as the elastostatic models obtained by means of the proposed technique predict more than 93.7% of the compliance deviations of a real industrial robot. The proposed method is simple enough to be jointly applicable to the most recent elastostatic model reduction techniques.

Linear equation systems for structural analysis: imagining resolutions

This work is based on a study into new ways of resolving the equilibrium equation systems for manual analyses of certain structures commonly found in building. It suggests finding solutions based on images that reproduce the operations of current methods, which may inspire the design of others that qualitatively reflect those of other more effective procedures. To date three methods (Gauss, Cholesky & Crout) have been imagined: (i) by “visualising” their operations through the mechanical behaviour of models during the equilibrium phase. These visualisations may help suggest other physical responses that can balance models more quickly and identify with new, more direct numerical methods; (ii) by “geometrising” operations by means of lines sketched freehand. This geometrisation may reveal hidden links between the parts of the calculation of current methods that enable more direct but equally precise new methods to be created. The paper shows four images to reinforce these viewpoints. Two visualise the methods of Gauss-Jordan and Cramer, confirming that the abstract procedures that resolve the systems may be linked to specific mechanical behaviours. The other two geometrise the resolutions by Gauss and Gauss-Jordan when the stiffness matrices are asymmetric. Their systems could emerge from the analysis of cracked models or from obtaining the equivalent actions in the P-Δ method, in line with a procedure drawn up previously. The paper ends by geometrising the resolution of a system at different scales and comparing the outcomes with those of numerical methods. The results (i) confirm that geometrising scalar and vectorial magnitudes for numerical analysis procedures reduces application times if they are calculated freehand; and (ii) point to possible lines of research for developing further graphic methods that can analyse other types of structure directly and accurately.

Optimization of a Class of n-Sub-Step Time Integration Methods for Structural Dynamics

This paper develops a family of optimized [Formula: see text]-sub-step time integration methods for structural dynamics, in which the generalized trapezoidal rule is used in the first [Formula: see text] sub-steps, and the last sub-step employs [Formula: see text]-point backward difference formula. The proposed methods can achieve second-order accuracy and unconditional stability, and their degree of numerical dissipation can range from zero to one. Also, the proposed methods can achieve the identical effective stiffness matrices for all sub-steps, reducing computational costs in the analysis of linear systems. Using the spectral analysis, optimized algorithmic parameters are presented, ensuring that the proposed methods can accurately calculate different types of dynamic problems such as wave propagation, stiff and nonlinear systems. Besides, with the increase in the number of sub-steps, the accuracy of the proposed methods can be enhanced without extra workload compared with single-step methods. Numerical experiments show that the proposed methods perform better in different dynamic systems.

Numerical Methods for Calculating Component Modes for Geometric Mistuning Reduced-order Models

Geometric mistuning models formulated from a component mode synthesis methods often require the calculation of component modes, particularly constraint and fixed interface normal modes, during substructuring. For Integrally Bladed Rotors, these calculations are required for each sector. This paper proposes methods that reuse information garnered from solving the constraint modes of a single sector on the remaining sectors to reduce memory requirements and solution times. A mesh metamorphosis tool is used to ensure finite element models match geometry obtained from a 3D optical scanner. This tool also produces a common mesh pattern from sector-to-sector. This is exploited to produce common permutation matrices and symbolic factorizations of sector stiffness matrices that are proposed for reuse in solving subsequent constraint modes. Furthermore, a drop tolerance is introduced to remove small values during constraint mode calculation to reduce memory requirements. It is proposed to reuse this dropping pattern produced from a single sector on the remaining sectors. Approaches are then extended to a parallel processing scheme to propose effective matrix partitioning methods. Finally, information gathered during the constraint mode calculations are reused during the solution of the fixed interface normal modes to improve solution time. Results show reusing permutation matrices and symbolic factorizations from sector-to-sector improves solution time and introduces no error. Using a drop tolerance is shown to reduce storage requirements of a constraint mode matrix, while reusing the dropping pattern introduces minimal error. Similarly, reusing constraint mode information in calculating normal modes greatly improves the performance.

Progressive Collapse Safety Evaluation of Truss Structures Considering Material Plasticity

Theoretical or numerical progressive collapse analysis is necessary for important civil structures in case of unforeseen accidents. However, currently, most analytical research is carried out under the assumption of material elasticity for problem simplification, leading to the deviation of analysis results from actual situations. On this account, a progressive collapse analysis procedure for truss structures is proposed, based on the assumption of elastoplastic materials. A plastic importance coefficient was defined to express the importance of truss members in the entire system. The plastic deformations of members were involved in the construction of local and global stiffness matrices. The conceptual removal of a member was adopted, and the impact of the member loss on the truss system was quantified by bearing capacity coefficients, which were subsequently used to calculate the plastic importance coefficients. The member failure occurred when its bearing capacity arrived at the ultimate value, instead of the elastic limit. The extra bearing capacity was embodied by additional virtual loads. The progressive collapse analysis was performed by iterations until the truss became a geometrically unstable system. After that, the critical progressive collapse path inside the truss system was found according to the failure sequence of the members. Lastly, the proposed method was verified against both analytical and experimental truss structures. The critical progressive collapse path of the experimental truss was found by the failure sequence of damaged members. The experimental observation agreed well with the corresponding analytical scenario, proving the method feasibility.