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This paper aims to proposes and analyzes a novel recovery-based posteriori error estimator for the stationary natural-convection problem based on penalized finite element method.

The optimal error estimates of the penalty FEM are established by using the lower-order finite element pair P₁-P₀-P₁ which does not satisfy the discrete inf-sup condition. Besides, a new recovery type posteriori estimator in view of the gradient recovery and superconvergent theory to deal with the discontinuity of the gradient of numerical solution.

The stability, accuracy and efficiency of the proposed method are confirmed by several numerical investigations.

The provided reliability and efficiency analysis is shown that the true error can be effectively bounded by the recovery-based error estimator.

We introduce a second‐order accurate time‐marching pressure‐correction algorithm to accommodate weakly‐compressible highly‐viscous liquid flows at low Mach number. As the incompressible limit is approached (Ma ≈ 0), the consistency of the compressible scheme is highlighted in recovering equivalent incompressible solutions. In the viscous‐dominated regime of low Reynolds number (zone of interest), the algorithm treats the viscous part of the equations in a semi‐implicit form. Two discrete representations are proposed to interpolate density: a piecewise‐constant form with gradient recovery and a linear interpolation form, akin to that on pressure. Numerical performance is considered on a number of classical benchmark problems for highly viscous liquid flows to highlight consistency, accuracy and stability properties. Validation bears out the high quality of performance of both compressible flow implementations, at low to vanishing Mach number. Neither linear nor constant density interpolations schemes degrade the second‐order accuracy of the original incompressible fractional‐staged pressure‐correction scheme. The piecewise‐constant interpolation scheme is advocated as a viable method of choice, with its advantages of order retention, yet efficiency in implementation.

Gives a bibliographical review of the error estimates and adaptive finite element methods from the theoretical as well as the application point of view. The bibliography at the end contains 2,177 references to papers, conference proceedings and theses/dissertations dealing with the subjects that were published in 1990‐2000.

The gradient recovery method proposed by Zhu and Zienkiewicz for one‐dimensional problems and extended to two dimensions by Silvester and Omeragi? is generalized to three‐dimensional solutions based on rectangular prism (brick) elements. The extension is not obvious so its details are presented, and the method compared with conventional local smoothing and direct differentiation. Illustrative examples are given, with an extensive experimental study of error. The method is computationally cheap and provides better accuracy than conventional local smoothing, but its accuracy is position dependent.

The purpose of this paper is to develop a compressive sensing (CS) algorithm for noisy solder joint imagery compression and recovery. A fast gradient-based compressive sensing (FGbCS) approach is proposed based on the convex optimization. The proposed algorithm is able to improve performance in terms of peak signal noise ratio (PSNR) and computational cost.

Unlike traditional CS methods, the authors first transformed a noise solder joint image to a sparse signal by a discrete cosine transform (DCT), so that the reconstruction of noisy solder joint imagery is changed to a convex optimization problem. Then, a so-called gradient-based method is utilized for solving the problem. To improve the method efficiency, the authors assume the problem to be convex with the Lipschitz gradient through the replacement of an iteration parameter by the Lipschitz constant. Moreover, a FGbCS algorithm is proposed to recover the noisy solder joint imagery under different parameters.

Experiments reveal that the proposed algorithm can achieve better results on PNSR with fewer computational costs than classical algorithms like Orthogonal Matching Pursuit (OMP), Greedy Basis Pursuit (GBP), Subspace Pursuit (SP), Compressive Sampling Matching Pursuit (CoSaMP) and Iterative Re-weighted Least Squares (IRLS). Convergence of the proposed algorithm is with a faster rate O(k*k) instead of O(1/k).

This paper provides a novel methodology for the CS of noisy solder joint imagery, and the proposed algorithm can also be used in other imagery compression and recovery.

According to the CS theory, a sparse or compressible signal can be represented by a fewer number of bases than those required by the Nyquist theorem. The new development might provide some fundamental guidelines for noisy imagery compression and recovering.

The gradient recovery method proposed by Zhu and Zienkiewicz for one‐dimensional problems is generalized to two dimensions, using quadrilateral elements. Its performance is compared with that of conventional local smoothing techniques and of direct differentiation of the finite‐element solution, on finite‐element approximations to analytically known polynomial and transcendental functions on a quadrilateral second‐order finite‐element mesh. The new method appears to be reliable and more stable than local smoothing, and to provide better accuracy than direct differentiation, at low computational cost.

An h‐adaptive finite element code for solving coupled Navier‐Stokes and energy equations is used to solve the thermally driven cavity problem for Rayleigh numbers at which no steady state exists (greater than 1.9 × 10⁸). This problem is characterised by sharp thermal and flow boundary layers and highly advection dominated transport, which normally requires special algorithms, such as streamline upwinding, to achieve stable and smooth solutions. It will be shown that h‐adaptivity provides a suitable solution to both of these problems (sharp gradients and advection dominated transport). Adaptivity is also very effective in resolving the flow physics, characterised by unsteady internal waves, are calculated for three Rayleigh numbers; 2 × 10⁸, 3 × 10⁸ and 4 × 10⁸ using a Prandtl number of 0.71 and results are compared with other published results.

An h‐adaptive finite element code for solving coupled Navier‐Stokes and energy equations is used to solve the thermally driven cavity problem. The buoyancy forces are represented using the Boussinesq approximation. The problem is characterised by very thin boundary layers at high values of Rayleigh number (>10⁶). However, steady state solutions are achievable with adequate discretisation. This is where the auto‐adaptive finite element method provides a powerful means of achieving optimal solutions without having to pre‐define a mesh, which may be either inadequate or too expensive. Steady state and transient results are given for six different Rayleigh numbers in the range 10³ to 10⁸ for a Prandtl number of 0.71. The use of h‐adaptivity, based on a posteriori error estimation, is found to ensure a very accurate problem solution at a reasonable computational cost.

This paper aims to present a review on utilizing shape memory technology (SMT) for active assembly/disassembly, i.e. assembly/disassembly without physically touching.

The fundamentals behind the shape memory effect (SME) in materials, in particular shape memory alloys (SMAs) and polymers, which are the cornerstones of SMT, are introduced, together with the possible approaches to implement this effect in active assembly/disassembly. Example applications for not only active assembly/ disassembly, but also programmed active disassembly are presented.

The advantages of utilizing SMT over conventional assembly/disassembly techniques are identified.

The paper introduces the fundamentals behind the SME and the basic approaches to implement the SMT in not only active assembly/disassembly, but also programmed active assembly.

A combination of both time step adaptivity and spatial mesh adaptivity is presented for transient magneto‐quasistatic fields.

Error controlled time step adaptivity is achieved using an implicit integration scheme and the spatial mesh resolution is adapted in each time step in order to effectively resolve the appearing and disappearing local transient saturation effects and eddy current layers. Two spatial refinement strategies are considered, the red‐green refinement leading to a regular mesh and the red refinement leading to an irregular mesh. Numerical results for 2d nonlinear magneto‐dynamic problems are presented.

An algorithm is proposed which computes the solution of a transient magnetostatic problem given a user prescribed error tolerance for the time stepping and the spatial refinement. The red refinement leading to irregular meshes requires projection techniques in the iterative conjugate gradient solver. However, the algorithm with red‐green refinement turns out to perform faster since the projection is too expensive.

The combination of error controlled time stepping and spatial adaptivity is firstly established in electromagnetic field computation.