An extended multiscale finite element method (EMsFEM) is developed for solving the mechanical problems of heterogeneous materials in elasticity. The underlying idea of the method is to construct numerically the multiscale base functions to capture the small-scale features of the coarse elements in the multiscale finite element analysis. On the basis of our existing work for periodic truss materials, the construction methods of the base functions for continuum heterogeneous materials are systematically introduced. Numerical experiments show that the choice of boundary conditions for the construction of the base functions has a big influence on the accuracy of the multiscale solutions, thus, different kinds of boundary conditions are proposed. The efficiency and accuracy of the developed method are validated and the results with different boundary conditions are verified through extensive numerical examples with both periodic and random heterogeneous micro-structures. Also, a consistency test of the method is performed numerically. The results show that the EMsFEM can effectively obtain the macro response of the heterogeneous structures as well as the response in micro-scale, especially under the periodic boundary conditions.

Abstract. This work presents ground based ultra low frequency (ULF) electromagnetic field measurements in the frequency range 0.1–10 Hz from January 2007 to December 2008. In this time period a strong earthquake series hits the Wenchuan region with a main shock of magnitude MS = 8.0 on 12 May 2008. The Hebei ULF electromagnetic observation network includes eight observation stations in north China and the observation system named E-EM is employed to record the electric potential difference between two electrodes with an analog automatic real-time continuous pen recorder. First, weak electric signals appeared on 11 October 2007 at Ningjin station, most of which are with relative long periods ~0.4–3 s and unequal amplitudes ~0.5–20 mm. Then, similar signals appeared at Gaobeidian station at the end of October. Abnormal behavior with various time intervals appeared randomly and not every day. At the beginning of April 2008, one and a half months before the Wenchuan MS = 8.0 earthquake, the anomalies were gradually subject to an intensive increase mainly in Gaobeidian SN direction and Ningjin EW direction. The abnormal behavior appeared almost every day and the amplitudes of electric signals, with short periods of ~0.1–0.3 s, enhanced to ~3–30 mm. Qingxian station started to record marginally high frequency signals in SN and EW components in the middle of April. On 9 May, 3 days before the main shock, the amplitude of high frequency information increased sharply at the same time in two components at Gaobeidian station and the maximum amplitude was up to 70 mm, i.e. 1.3 mV m−1 for the electric field. This situation did not stop until 17 May, 5 days after the main event. However, this kind of climax phenomena did not happen at Ningjin station and Qingxian station. Then weak anomalous information lasted about four months again, and strong signals appeared again for a short time before several powerful aftershocks. It is the first time that an abnormity with so large an amplitude and so long a duration time in the observation history of this network though several strong earthquakes were recorded. Furthermore, no obvious interferences have been found during this period. So this event is possibly related to this shock although all these three stations are more than 1300 km away from the Wenchuan earthquake epicenter.

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