We consider two mathematical problems that are connected and occur in the layer-wise production process of a workpiece using wire-arc additive manufacturing. As the first task, we consider the ...automatic construction of a honeycomb structure, given the boundary of a shape of interest. In doing this, we employ Lloyd’s algorithm in two different realizations. For computing the incorporated Voronoi tesselation we consider the use of a Delaunay triangulation or alternatively, the eikonal equation. We compare and modify these approaches with the aim of combining their respective advantages. Then in the second task, to find an optimal tool path guaranteeing minimal production time and high quality of the workpiece, a mixed-integer linear programming problem is derived. The model takes thermal conduction and radiation during the process into account and aims to minimize temperature gradients inside the material. Its solvability for standard mixed-integer solvers is demonstrated on several test-instances. The results are compared with manufactured workpieces.
One crucial advantage of additive manufacturing regarding the optimization of lattice structures is that there is a reduction in manufacturing constraints compared to classical manufacturing methods. ...To make full use of these advantages and to exploit the resulting potential, it is necessary that lattice structures are designed using optimization. Against this backdrop, two mixed integer programs are developed in order to use the methods of mathematical optimization in the context of topology optimization on the basis of a fitted ground structure method. In addition, an algorithm driven product design process is presented to systematically combine the areas of mathematical optimization, computer aided design, finite element analysis and additive manufacturing. Our developed computer aided design tool serves as an interface between state-of-the-art mathematical solvers and computer aided design software and is used for the generation of design data based on optimization results. The first mixed integer program focuses on powder-based additive manufacturing, including a preprocessing that allows a multi-material topology optimization. The second mixed integer program generates support-free lattice structures for additive manufacturing processes usually depending on support structures, by considering geometry-based design rules for inclined and support-free cylinders and assumptions for location and orientation of parts within a build volume. The problem to strengthen a lattice structure by local thickening or beam addition or both, with the objective function to minimize costs, is modeled. In doing so, post-processing is excluded. An optimization of a static area load with a practice-oriented number of connection nodes and beams was manufactured using the powder-based additive manufacturing system EOS INT P760.
Three-dimensional (3D) printing, also known as additive manufacturing (AM), has emerged in the last decades as an innovative technology to build complex structures. It enables increasing design ...complexity and low-cost customization with a vast range of materials. AM capabilities contributed to a widespread acceptance of 3D printing in different industries such as the aerospace and the automotive. However, important issues and limitations still need to be addressed, namely in printing complex objects where supports and material roughness surface are to be minimized. In this work we consider a 5–axis printer with the three traditional
xyz
movements and two additional degrees of freedom on the printer table bed. These extra degrees of freedom (table bed rotation and tilt) allow printing of more complex objects, and we propose an approach which consists on the decomposition of complex objects into simpler parts, allowing each part to be printed in an optimal way. We aim to reduce the number of supports needed and attain high final object quality due to lower material surface roughness. The optimal printing direction (or, equivalently, rotation) and sequencing of the object parts is determined by solving a combinatorial sequencing optimization problem. All the local or global optimal parts rotations are obtained by solving a global optimization sub-problem for each part, and are taken as input parameters for the sequencing optimization problem. We provide a heuristic approach for the combinatorial sequencing optimization problem, and a multistart multisplit search methodology for computing all the local or global optimal parts rotations for the sub-problems.
Many finance questions require the predictive distribution of returns. We propose a bivariate model of returns and realized volatility (RV), and explore which features of that time-series model ...contribute to superior density forecasts over horizons of 1 to 60 days out of sample. This term structure of density forecasts is used to investigate the importance of: the intraday information embodied in the daily RV estimates; the functional form for
log
(
R
V
)
dynamics; the timing of information availability; and the assumed distributions of both return and
log
(
R
V
)
innovations. We find that a joint model of returns and volatility that features two components for
log
(
R
V
)
provides a good fit to S&P 500 and IBM data, and is a significant improvement over an EGARCH model estimated from daily returns.
This paper presents an adapted trust-region method for computationally expensive black-box optimization problems with mixed binary variables that involve a cyclic symmetry property. Mixed binary ...problems occur in several practical optimal design problems, e.g., aircraft engine turbines, mooring lines of offshore wind turbines, electric engine stators and rotors. The motivating application for this study is the optimal design of helicopter bladed disk turbomachines. The
necklace
concept is introduced to deal with the cyclic symmetry property, and to avoid costly black-box objective-function evaluations at equivalent solutions. An adapted distance is proposed for the discrete-space exploration step of the optimization method. A convergence analysis is presented for the trust-region derivative-free algorithm, DFOb-
d
H
, extended to the mixed-binary case and based on the Hamming distance. The convergence proof is extended to the new algorithm, DFOb-
d
neck
, which is based on the necklace distance. Computational comparison with state-of-the-art black-box optimization methods is performed on a set of analytical problems and on a simplified industrial application.
This note is concerned with inverse wave scattering in one and two dimensional domains. It seeks to recover an unknown function based on measurements collected at the boundary of the domain. For ...one-dimensional problem, only one point of the domain is assumed to be accessible. For the two dimensional domain, the outer boundary is assumed to be accessible. It develops two iterative algorithms, in which an assumed initial guess for the unknown function is updated. The first method uses a set of sampling functions to formulate a
moment problem
for the correction to the assumed value. This method is applied to both one-dimensional and two dimensional domains. For two dimensional Helmholtz equation, it relies on a new effective filtering technique which is another contribution of the present work. The second method uses a direct formulation to recover the correction term. This method is only developed for the one-dimensional case. For all cases presented here, the correction to the assumed value is obtained by solving an over-determined linear system through the use of least-square minimization. Tikhonov regularization is also used to stabilize the least-square solution. A number of numerical examples are used to show their applicability and robustness to noise.
We investigate
C
1
finite element methods for one dimensional elliptic distributed optimal control problems with pointwise constraints on the derivative of the state formulated as fourth order ...variational inequalities for the state variable. For the problem with Dirichlet boundary conditions, we use an existing
H
5
2
-
ϵ
regularity result for the optimal state to derive
O
(
h
1
2
-
ϵ
)
convergence for the approximation of the optimal state in the
H
2
norm. For the problem with mixed Dirichlet and Neumann boundary conditions, we show that the optimal state belongs to
H
3
under appropriate assumptions on the data and obtain
O
(
h
) convergence for the approximation of the optimal state in the
H
2
norm.
Motivated by problems arising in magnetic drug targeting, we propose to generate an almost constant Kelvin (magnetic) force in a target subdomain, moving along a prescribed trajectory. This is ...carried out by solving a minimization problem with a tracking type cost functional. The magnetic sources are assumed to be dipoles and the control variables are the magnetic field intensity, the source location and the magnetic field direction. The resulting magnetic field is shown to effectively steer the drug concentration, governed by a drift-diffusion PDE, from an initial to a desired location with limited spreading.
New algorithms are developed to adapt the convergence tolerances for the constraint and adjoint equations for practical engineering optimization problems solved using the discrete adjoint approach. ...The algorithms are designed to achieve design order convergence of the optimization algorithm at reduced computational cost. We have found based on analysis and supported by numerical experimentation that adapting both the constraint and adjoint equation tolerances based on the norm of the gradient is sufficient to achieve design order convergence. We have also found that adapting the constraint equation tolerance is necessary, though we were not able to show analytically that adapting the adjoint equation tolerance is necessary. Based on the numerical experimentation, it appears that design order convergence can be achieved without adapting the adjoint equation in some cases but not others. The gain in computational efficiency using the new algorithms over using fixed tolerances is demonstrated through three numerical test problems.
In this work, metamodel-based robust optimization is performed using measured scatter of noise variables. Principal component analysis is used to describe the input noise using linearly uncorrelated ...principal components. Some of these principal components follow a normal probability distribution, others however deviate from a normal probability distribution. In that case, for more accurate description of material scatter, a multimodal distribution is used. An analytical method is implemented to propagate the noise distribution via metamodel and to calculate the statistics of the response accurately and efficiently. The robust optimization criterion as well as the constraints evaluation are adjusted to properly deal with multimodal response. Two problems are presented to show the effectiveness of the proposed approach and to validate the method. A basketball free throw in windy weather condition and forming of B-pillar component are presented. The significance of accounting for non-normal distribution of input variables using multimodal distributions is investigated. Moreover, analytical calculation of response statistics, and adjustment of the robust optimization problem are presented and discussed.
PDF
