In bulk metal forming, an initially simple part is plastically deformed between dies and tools to obtain a desired final shape. In this manner near-net shape products can be obtained with a minimal material waste in advantageous production cycles. Also in the case of cold forging, final products with excellent dimensional accuracy and mechanical properties can be obtained. Such benefits have led metal forming parts to be widely used for a variety of structural parts and mechanical components.
Recently, as near net-shape manufacturing has become a necessity in industry to meet the demands of reducing cost and improving quality, efforts are being made to replace traditional experience-based process development with more effective and efficient design methods. Among the numerous numerical methods that can be used for the analysis of bulk metal forming processes, the finite element method (FEM) is most widely used today due to its accuracy and capability of dealing with general shaped geometries. In particular, the rigid-viscoplastic finite element approach is preferred over other approaches due to its computational efficiency. Furthermore, due to geometrical constraints of two-dimensional analyses, emphasis on three-dimensional analyses is ever growing. However, the large amount of computational effort and expense required for three-dimensional analyses can limit their actual use in bulk forming industry. Thus, efforts are needed to make three-dimensional simulations more attractive.
In terms of hardware, the use of lower cost computing machines, such as personal computers (PCs), is suitable for the bulk metal forming industry, which is largely composed of small-to-medium sized enterprises. In terms of software, the application of parallel processing algorithms to bulk forming simulations is promising, as parallel processing distributes computation time and memory requirements among a number of processors to increase the efficiency of calculations. In this regard, PC clusters, a group of PCs connected by local area network (LAN) for communication, can be highly beneficial for bulk forming simulations, since PC clusters maintain the benefit of inexpensive computing hardware while providing parallel processing capability that can increase the efficiency of numerical simulations.
Parallel processing reduces computing time by distributing the required computing tasks among processors such that computations can be carried out concurrently. Ideally the computing time needed for a sequential run should be reduced by a factor of p, where p is the total number of processors used in the parallel run. However, such is not the case in reality. According to Amdahl’s law, the sequential portion of an algorithm theoretically limits the gain in speed-up that can be obtained. Also, parallel computing overhead, which includes cost for communication between processors and synchronization of computations among processors, acts to diminish parallel computing performance. This means that there is a limit to the time saving benefit that can be attained regardless of the parallel algorithm used.
Thus, it can be of great benefit if the number of processors that will give the best parallel computing performance for a given parallel algorithm is known a priori. In addition, the relative performance of various parallel algorithms will likely depend on the given bulk forming problem. The main objective of the current study is the development of a pre-simulation tool in a PC cluster environment that can guide in selection of an appropriate parallel algorithm and the number of processors to be used for a given bulk forming problem. Also, considering the two types of CPUs that compose the current PC cluster, the computing capability of both sets of CPUs was first investigated.
Two parallel algorithms using direct solving techniques, denoted as the column and sub-domain approaches, respectively, were currently investigated. In the column approach, columns of the assembled global stiffness matrix were distributed amongst processors and parallel $LDL^T$ factorization was used to solve the global matrix equation. In the sub-domain approach, domain decomposition was used to divide physical domain of the problem into smaller sub-domains. This enabled solving the reduced form of the global stiffness matrix equation. The first approach has the advantage of using a simpler algorithm, while the second method has the benefit of dealing with smaller sub-domains independently per processor.
The current pre-simulation tool was based on operation counts and time estimates. The exact computation time of a parallel program depends on numerous factors, but for simplicity, the numerical operation counts were used to make rough estimates of computing times in the present study. The operation counts of major computing routines in both the column and sub-domain approaches were determined. Then time coefficients for the operation counts were determined through numerical experiments of various upsetting simulations. The pre-simulation tool compared the time estimates of the two approaches for a given bulk forming problem to select the method of parallel solving and the number of processors to be used that would give the faster computation.
The usefulness of the pre-simulation tool was examined through various practical bulk forming simulation examples, such as forging of swash plate, spur gear, connecting rod, crankshaft, constant velocity joint, and spider. It was found that the computing times for these forging simulations varied depending on the parallel algorithm and number of processors used. For the spur gear, crankshaft, constant velocity joint, and spider simulations, it was found that the column approach gave better results. On the other hand, the sub-domain approach was better for the swash plate and connecting rod cases. The developed pre-simulation tool was shown to successfully recommend the algorithm and number of processors for each simulation example to reduce computing time. Considering the economical benefit of PC clusters, it is construed that the results of the current study can lead to more effective use of three-dimensional FE simulations in the design of bulk forming processes.
부피성형 공정은 단순한 형태의 소재를 금형 안에 삽입한 후 압축력을 가해 복잡한 형상의 제품으로 성형하는 공정이다. 부피성형 공정은 일체형 제품의 대량 생산이 가능하여 생산 단가를 낮출 수 있으며 가공 경화 등의 효과로 인해 기계적 성질이 우수하다. 3차원 강점소성 유한요소해석은 부피성형 제품의 품질 향상 및 공정설계의 효율성 증대에 큰 도움이 될 수 있다. 그러나 3 차원 해석이 요구하는 막대한 컴퓨팅 시간 및 메모리 문제 때문에 아직 산업체에서 널리 사용되지 못하는 것이 실정이다.
PC 클러스터는 여러 PC를 랜으로 연결하여 사용하는 컴퓨팅 환경으로서, 저가인 PC의 이점과 계산의 효율을 증대 시킬 수 있는 병렬 컴퓨팅 능력을 동시에 가지고 있다. 일반적으로 병렬처리는 계산량을 여러 프로세서로 분배 시켜 계산을 동시에 수행함으로 효율성을 증대 시키는 방법이다. 그러나 Amdahl의 법칙 및 병렬 컴퓨팅의 코스트로 인해 병렬처리 기법으로 얻을 수 있는 효율성 향상에는 한계가 있다. 따라서, 주어진 문제에 가장 적합한 병렬 알고리즘 및 사용할 프로세서 수를 미리 알 수 있으면 보다 효과적으로 병렬 기법을 3 차원 부피성형해석에 적용시킬 수 있을 것이다.
본 연구에서는 PC 클러스터 환경에서 효과적인 병렬처리를 수행하기 위한 예비해석 도구의 개발을 목표로 하였다. 개발된 예비해석 도구는 두 병렬 알고리즘을 고려하고 있다. 첫 알고리즘은 유한요소해석에서 얻어지는 강성행렬의 행을 여러 프로세서에 할당하여 문제를 접근하는 방법이고, 두 번째 방법은 유한요소해석의 전체 영역을 여러 개의 부영역으로 나누어서 문제를 푸는 방법이다. 이와 같은 두 병렬 알고리즘은 각각 장단점을 갖고 있기 때문에 주어진 부피성형 문제에 따라 계산 단축 효과가 다르다. 또한 사용하기 적합한 프로세서의 수도 주어진 문제와 알고리즘에 따라 다르다.
본 연구에서 개발된 예비해석 도구는 주어진 부피성형 해석예제를 대상으로 사용하기 적합한 알고리즘과 프로세서 수를 제시는 것을 목표로 하였다. 이를 위해 각 알고리즘이 요구하는 대표적인 연산 수를 분석하였고, 이러한 연산이 요구하는 계산시간을 수치적인 실험을 통해 평가하였다. 이러한 과정을 통해 주어진 부피성형 문제가 요구하는 계산시간을 대략적으로 예측할 수 있었으며, 이와 같이 예측되는 계산시간을 비교하여 주어진 문제에 적합한 병렬 알고리즘 및 프로세서의 수를 예비해석 도구가 제안 하는 것이다.
개발된 예비해석 도구의 유용성을 평가하기 위하여 이를 다양한 부피성형 해석 예제에 적용을 시켜보았다. 스와시 플레이트, 스퍼 기어, 커넥팅 로드, 크랭크쉐프트, CVT, 그리고 스파이더 부품의 3차원 성형공정 해석을 수행하였다. 예비해석 도구가 각 예제에 적합한 알고리즘 및 프로세서 수를 제안하는 것을 확인 할 수 있었다. 스퍼 기어, 크랭크쉐프트, CVT, 그리고 스파이더의 경우에는 강성행렬의 행을 나누어서 푸는 병렬기법이 효과적이었으며, 스와시 플레이트와 커넥팅 로드의 예제는 영역분할법이 더 좋은 결과를 보였다. 이러한 예비해석 도구는 병렬 3 차원 부피성형 해석의 효과적인 사용에 도움을 줄 것으로 기대한다.
