
Dai, K.Y.; Liu, Guirong
In this paper, the smoothed finite element method (SFEM) is proposed for 2D elastic problems by incorporation of the cellwise strain smoothing operation into the conventional finite elements. When a constant smoothing function is chosen, area integration becomes line integration along cell boundaries and no derivative of shape functions is
needed in computing the field gradients. Both static and dynamic numerical examples are analyzed in the paper. Compared with the conventional FEM, the SFEM achieves more accurate results and generally higher convergence rate in energy without increasing computational cost. In addition, as no mapping or coordinate transformation is performed in
the SFEM,...

Tan, Alex Y.K.; Patera, Anthony T.
We solve the 2nd order wave equation, hyperbolic and linear in nature, for the pressure distribution of onedimensional
seismic problem with smooth initial pressure and rate of pressure change. The reduced basis method, offlineonline computational procedures and a posteriori error estimation are developed. We show that the reduced basis pressure distribution is an accurate approximation to the finite element pressure
distribution and the offlineonline computational procedures work well. The a posteriori error estimation developed shows
that the ratio of the maximum error bound over the maximum norm of the reduced basis error has a constant magnitude of O(10²). The inverse problem works well,...

Huynh, Dinh Bao Phuong; Peraire, Jaime; Patera, Anthony T.; Liu, Guirong
This paper reports the development of reduced
basis approximations, rigorous a posteriori error bounds, and offlineonline computational procedures for the accurate, fast and reliable predictions of stress intensity factors or strain energy release rate for “Mode I” linear elastic crack problem. We demonstrate the efficiency and rigor of our numerical method in several examples. We apply our method to a practical failure design application.

Li, Zirui; Liu, Guirong; Chen, Yuzong; Wang, JianSheng; Hadjiconstantinou, Nicolas; Cheng, Y.; Han, J.
Linear thermodynamics transportation theory is employed to study filtration of rodlike DNA molecules. Using the repeated nanoarray consisting of alternate deep and shallow regions, it is demonstrated that the complex partitioning of rodlike DNA molecules of different lengths can be described by traditional transport theory with the configurational entropy properly quantified. Unlike most studies at mesoscopic level, this theory focuses on the macroscopic group behavior of DNA transportation. It is therefore easier to conduct validation analysis through comparison with experimental results. It is also promising in design and optimization of DNA filtration devices through
computer simulation.

Nguyen, N. C.; Peraire, Jaime
In the presence of nonaffine and highly nonlinear terms in parametrized partial differential equations, the standard Galerkin reducedorder approach is no longer efficient, because the evaluation of these terms involves high computational complexity. An efficient reducedorder approach is developed to deal with “nonaffineness” and nonlinearity. The efficiency and accuracy of the approach are demonstrated on several test cases, which show significant computational savings relative to classical numerical methods and relative to the standard Galerkin reducedorder approach.

Ali, S.; Damodaran, Murali; Willcox, Karen E.
Many industrial applications in engineering and science are solved using commercial engineering solvers that function as blackbox simulation tools. Besides being time consuming and computationally expensive, the actual mathematical models and underlying structure of these problems are for most part unknown. This paper presents a method for constructing approximate lowdimensional models for such problems using the proper orthogonal decomposition (POD) technique. We consider a heat diffusion problem and a contamination transport problem, where the actual mathematical models are assumed to be unknown but numerical data in time are available so as to enable the formation of an ensemble of snapshots...