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.

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. Additionally, it is shown that reusing the same dropping pattern introduces minimal error without degradation in solution times. Similarly, reusing the information from constraint modes for calculating fixed interface normal modes greatly improves the performance in a shift-and-invert technique for solving eigenvalue problems.

Sparse matrix partitioning for optimizing SpMV on CPU-GPU heterogeneous platforms

Sparse matrix–vector multiplication (SpMV) kernel dominates the computing cost in numerous applications. Most of the existing studies dedicated to improving this kernel have been targeting just one type of processing units, mainly multicore CPUs or graphics processing units (GPUs), and have not explored the potential of the recent, rapidly emerging, CPU-GPU heterogeneous platforms. To take full advantage of these heterogeneous systems, the input sparse matrix has to be partitioned on different available processing units. The partitioning problem is more challenging with the existence of many sparse formats whose performances depend both on the sparsity of the input matrix and the used hardware. Thus, the best performance does not only depend on how to partition the input sparse matrix but also on which sparse format to use for each partition. To address this challenge, we propose in this article a new CPU-GPU heterogeneous method for computing the SpMV kernel that combines between different sparse formats to achieve better performance and better utilization of CPU-GPU heterogeneous platforms. The proposed solution horizontally partitions the input matrix into multiple block-rows and predicts their best sparse formats using machine learning-based performance models. A mapping algorithm is then used to assign the block-rows to the CPU and GPU(s) available in the system. Our experimental results using real-world large unstructured sparse matrices on two different machines show a noticeable performance improvement.

A Numerical Method for Flexural Vibration Band Gaps in A Phononic Crystal Beam with Locally Resonant Oscillators

The differential quadrature method has been developed to calculate the elastic band gaps from the Bragg reflection mechanism in periodic structures efficiently and accurately. However, there have been no reports that this method has been successfully used to calculate the band gaps of locally resonant structures. This is because, in the process of using this method to calculate the band gaps of locally resonant structures, the non-linear term of frequency exists in the matrix equation, which makes it impossible to solve the dispersion relationship by using the conventional matrix-partitioning method. Hence, an accurate and efficient numerical method is proposed to calculate the flexural band gap of a locally resonant beam, with the aim of improving the calculation accuracy and computational efficiency. The proposed method is based on the differential quadrature method, an unconventional matrix-partitioning method, and a variable substitution method. A convergence study and validation indicate that the method has a fast convergence rate and good accuracy. In addition, compared with the plane wave expansion method and the finite element method, the present method demonstrates high accuracy and computational efficiency. Moreover, the parametric analysis shows that the width of the 1st band gap can be widened by increasing the mass ratio or the stiffness ratio or decreasing the lattice constant. One can decrease the lower edge of the 1st band gap by increasing the mass ratio or decreasing the stiffness ratio. The band gap frequency range calculated by the Timoshenko beam theory is lower than that calculated by the Euler-Bernoulli beam theory. The research results in this paper may provide a reference for the vibration reduction of beams in mechanical or civil engineering fields.