The aim of the present paper is to evaluate the use of linear and quadratic approximating response surfaces as metamodels in a reliability assessment of a sheet metal forming process using the Monte Carlo simulation technique. Monte Carlo simulation was used to determine the probability for springback and thickness variation in a sheet metal part. The conclusions of this study is that Monte Carlo analysis can be used to identify the most important variables and to estimate the range of the studied responses. Linear metamodels can be used to identify the important variables and to give an estimate of the probabilistic response. But quadratic surfaces are required for a more accurate analysis.

In the present paper, an optimization technique has been used to minimize the risk of failure in a sheet metal forming process. Two different types of finite element solvers, one using total plasticity and the other using incremental plasticity, have been used. A comparison between response surface methodology and space mapping (SM) with the one-step solver as surrogate model has been done. The conclusion of this study is that the use of the total plasticity theory drastically reduces the required computing time. Furthermore, the solution from the SM optimization algorithm is close to the solution obtained by the incremental plasticity solver.

In the present paper optimization has been used to evaluate alternative sheet metal forming processes. Six process set-ups were first defined in a hierarchy of designs and optimization was then used to evaluate each forming process of these designs. The challenge in designing the forming process was to avoid failure in the material and at the same time reach an acceptable through thickness strain. The conclusions of this study is that there may exist a different process that can give an improved product for the desired geometry. This process might be impossible for the optimization algorithm to reach due to either a poor starting point or a not so wise process set-up.

The longitudinal movement of blood vessel walls has so far gained little or no attention, as it has been presumed that these movements are of a negligible magnitude. However, modern high-resolution ultrasound scanners can demonstrate that the inner layers of the arterial wall exhibit considerable movements in the longitudinal direction. This paper evaluates a new, noninvasive, echo-tracking technique, which simultaneously can track both the radial and the longitudinal movements of the arterial wall with high resolution in vivo. Initially, the method is evaluated in vitro using a specially designed ultrasound phantom, which is attached to and moved by an X-Y system, the movement of which was compared with two high-resolution triangulation lasers. The results show an inaccuracy of 2.5% full scale deflection (fsd), reproducibility of 12 µm and a resolution of 5 µm, which should be more than sufficient for in vivo studies. The ability of the method is also demonstrated in a limited in vivo study in which a preselected part of the inner vessel wall of the right common carotid artery of a healthy volunteer is tracked in two dimensions over many cardiac cycles. The results show well reproducible x-y movement loops in which the recorded radial and longitudinal movements both are of the magnitude millimetre. © 2005 IEEE.

In the present paper, an optimization of the draw-in of an automotive sheet metal part has been carried out using response surface methodology (RSM) and space mapping technique. The optimization adjusts the draw bead restraining force in the model such that the draw-in in the FE-model corresponds to the draw-in in the physical process. The conclusion of this study is that space mapping is a very effective and accurate method to use when calibrating the draw-in of a sheet metal process. In order to establish draw bead geometry from the draw bead restraining force a 2D-model was utilized. The draw bead geometry found showed good agreement with the physical draw bead geometry.

The potential of using simulation and optimization techniques in the design of sheet metal forming processes has been investigated. Optimization has been used in a variety of sheet metal forming applications. This usage has given ideas and guidelines on how to formulate an optimization problems, how to ensure its rapid convergence and how to reach a feasible process design. Furthermore one method to include the uncertainty and variation in the governing parameters has been evaluated.

For each forming process many formability problems exist which usually are associated with fracture, wrinkling and springback. The aim of a good process design is to avoid these problems. By the use of simulation based design the physical trial- and error iterations can to a large extent be replaced by virtual iterations. When optimization techniques are used in the design process the number of iterations can be very large. In this work effort has been made to develop an optimization method, Space Mapping (SM), which is less computing intensive. It has been shown that SM can drastically reduce the required computing time and that its optimal solution is close to the optimal solution obtained by the more computing intensive Response Surface Methodology (RSM). The drawback of SM is that the method is less robust compared to RSM, and that it requires a better initial design for convergence.

Each industrial sheet metal forming process exhibits a certain degree of stochastic behavior due to uncertainties and variations in material properties, geometry and other process parameters. The forming process must be designed such that it is insensitive to these uncertainties and variations, i.e. the process must be robust. One possibility to consider uncer tainties and variations in a simulation and optimization based design process is to use the Monte Carlo method, in particular in combination with response surfaces as metamodels. It has been shown in this work that a linear response surface can successfully be used to identify the important design variables and to give an estimate of the probabilistic response. It has also been found that quadratic surfaces are required for a more accurate evaluation of the response.

The aim of this paper is to determine if the Space Mapping technique using surrogate models together with response surfaces is useful in the optimization of crashworthiness and sheet metal forming. In addition, the efficiency of optimization using Space Mapping will be compared to traditional structural optimization using the Response Surface Methodology (RSM). Five examples are used to study the algorithm: one optimization of an analytic function and four structural optimization problems. All examples are constrained optimization problems. In all examples, the algorithm converged to an improved design with all constraints fulfilled, even when a conventional RSM optimization failed to converge. For the crashworthiness design problems, the total computing time for convergence was reduced by 53% using Space Mapping compared to conventional RSM. For the sheet metal forming problems the total computing time was reduced by 63%. The conclusions are that optimization using Space Mapping and surrogate models can be used for optimization in crashworthiness design and sheet metal forming applications with a significant reduction in computing time.

The aim of the present work is to find a stable and effective optimization algorithm that can be used to determine the location and size of drawbeads in sheet metal forming processes. The result from the optimization will be the restraining force that each drawbead applies to the blank. In addition, optimization has been used to determine the physical dimensions of the drawbead.

The study in Paper I is limited to two structures in crashworthiness design, each with two design variables and three responses. The conclusion is that 1.5 times the minimum number of function evaluation should be used to fit the surface approximations. The conclusion of this study gives an indication on how many evaluations that are needed in order to efficiently construct the response surface approximations.

Paper II shows that the optimization algorithm using Space Mapping is well suited for optimization problems in crashworthiness design and in the design of sheet metal forming processes. All optimization applications converged to the correct optimum and the computing time was decreased with a maximum of 63% relative to the traditional RSM optimization.

The Space Mapping algorithm presented in Paper III converged to an optimum value that is lower than the optimum value from the traditional RSM, when the approximated surfaces from the first RSM iteration were used as the coarse model. The Space Mapping algorithm, however, reached a higher value compared to RSM, when surfaces from the third RSM iteration was used as the coarse model. Hence, the Space Mapping algorithm converged to a better objective value compared to RSM with less evaluations.

This thesis has shown that it is possible to apply optimization on the design of sheet metal forming processes. For the problem used in the last paper both the traditional Response Surface Methodology and the Space Mapping technique were successful in avoiding failure due to necking and to decrease the risk of wrinkles. The Response Surface Methodology is robust enough to be used in the industrial tool design process. The Space Mapping technique using linear response surfaces as coarse model needs further research before it can be used efficiently in industry.