快速判定几何约束奇异性的切面扰动法

针对冗余奇异和分支奇异的判定问题,提出一种新的切面扰动的判定方法.该方法将奇异的雅可比矩阵分为独立构型空间和奇异空间,变量沿独立构型空间的切面扰动,计算更新的雅克比矩阵的秩,依据秩亏的变化可以快速、稳定地判定约束奇异性.该算法克服了残量扰动法的数值迭代、计算量大和不稳定的缺点,并且在参数化特征造型系统InteSolid中得到验证.     

Matrix-free aerostructural optimization of aircraft wings

In structural optimization subject to failure constraints, computing the gradients of a large number of functions with respect to a large number of design variables may not be computationally practical. Often, the number of constraints in these optimization problems is reduced using constraint aggregation at the expense of a higher mass of the optimal structural design. This work presents results of structural and coupled aerodynamic and structural design optimization of aircraft wings using a novel matrix-free augmented Lagrangian optimizer. By using a matrix-free optimizer, the computation of the full constraint Jacobian at each iteration is replaced by the computation of a small number of Jacobian-vector products. The low cost of the Jacobian-vector products allows optimization problems with thousands of failure constraints to be solved directly, mitigating the effects of constraint aggregation. The results indicate that the matrix-free optimizer reduces the computational work of solving the optimization problem by an order of magnitude compared to a traditional sequential quadratic programming optimizer. Furthermore, the use of a matrix-free optimizer makes the solution of large multidisciplinary design problems, in which gradient information must be obtained through iterative methods, computationally tractable.     

Mixed Global Constraints and Inference in Hybrid CLP–IP Solvers

The complementing strengths of Constraint (Logic) Programming (CLP) and Mixed Integer Programming (IP) have recently received significant attention. Although various optimization and constraint programming packages at a first glance seem to support mixed models, the modeling and solution techniques encapsulated are still rudimentary. Apart from exchanging bounds for variables and objective, little is known of what constitutes a good hybrid model and how a hybrid solver can utilize the complementary strengths of inference and relaxations. This paper adds to the field by identifying constraints as the essential link between CLP and IP and introduces an algorithm for bidirectional inference through these constraints. Together with new search strategies for hybrid solvers and cut-generating mixed global constraints, solution speed is improved over both traditional IP codes and newer mixed solvers.     

Implementation of set-based design in multidisciplinary design optimization

Set-based design is a design approach where feasible regions for the design variables are determined from different disciplines, with the goal of locating and working with the areas of feasible overlap. During the process the constraints are adjusted in order to accommodate conflicting requirements between disciplines. The main objective of set-based design is to narrow the design space, while delaying the pursuit of a single point design as much as possible. This process avoids finalizing decisions early and allows for flexibility in dealing with requirement creep. This paper presents the development and application of a new multidisciplinary design optimization (MDO) algorithm inspired by the principles of set-based design. The new MDO algorithm was developed with the core concept of describing the design using sets to incorporate features of set-based design and achieve greater flexibility than with a single-point optimization. The MDO algorithm was applied to a ship design problem and the ship design application demonstrated the value of utilizing set-based design as a space-reducing technique before approaching the problem with a point-based optimization. Furthermore, incorporating flexibility in the constraints allowed the optimization to handle a problem with very strict constraints in a rational manner and minimize the necessary constraint violation.     

雅可比矩阵引导的无翻转体映射生成

目的体映射建立了两个3维体网格之间的对应关系,是计算机图形学中的重要研究方向。很多应用要求体映射是无翻转的,即其雅可比矩阵的行列式处处大于0。然而,现有的无翻转体映射生成算法经常无法完全消除翻转。挑战主要在于很难在保证满足位置约束的前提下消除映射的翻转。为此,提出一种新的无翻转体映射计算方法,核心是一种新的变形方法。方法首先放松位置约束,然后在变形过程中通过线搜索的方式保证不产生翻转,最后将网格无翻转地变形到满足位置约束。为实现这个变形过程,提出一种雅可比矩阵引导的变形算法。虽然现有的无翻转体映射方法不能完全消除翻转,但其雅可比矩阵可以作为本文变形算法的指导。此外,优化了位置能量,使得变形网格最终能够满足位置约束要求。为了满足体映射低扭曲的要求,算法最后在固定位置约束的前提下进一步优化了体映射的扭曲能量。结果对大量复杂网格进行实验,本文算法能够保证生成无翻转的体映射,并且通过多步优化最终结果均能满足给定的位置约束要求。结论通过与现有其他算法的优点和局限性对比,结果表明本文算法具有较好的鲁棒性。本文算法从一个全新的角度促进了无翻转体映射生成技术的进步与发展。     

Multi-way versus one-way constraints in user interfaces: Experience with the deltablue algorithm

The efficient satisfaction of constraints is essential to the performance of constraint-based user interfaces. In the past, most constraint-based user interfaces have used one-way rather than multi-way constraints because of a widespread belief that one-way constraints were more efficient. In this paper we argue that many user interface construction problems are handled more naturally and elegantly by multi-way constraints than by one-way constraints. We present pseudocode for an incremental multi-way constraint satisfaction algorithm, DeltaBlue, and describe experience in using the algorithm in two user interface toolkits. Finally, we provide performance figures demonstrating that multi-way constraint solvers can be entirely competitive in performance with one-way constraint solvers.     

View-based propagation of decomposable constraints

Constraints that may be obtained by composition from simpler constraints are present, in some way or another, in almost every constraint program. The decomposition of such constraints is a standard technique for obtaining an adequate propagation algorithm from a combination of propagators designed for simpler constraints. The decomposition approach is appealing in several ways. Firstly because creating a specific propagator for every constraint is clearly infeasible since the number of constraints is infinite. Secondly, because designing a propagation algorithm for complex constraints can be very challenging. Finally, reusing existing propagators allows to reduce the size of code to be developed and maintained. Traditionally, constraint solvers automatically decompose constraints into simpler ones using additional auxiliary variables and propagators, or expect the users to perform such decomposition themselves, eventually leading to the same propagation model. In this paper we explore views, an alternative way to create efficient propagators for such constraints in a modular, simple and correct way, which avoids the introduction of auxiliary variables and propagators.     

A full-space barrier method for stress-constrained discrete material design optimization

In this paper, we present a full-space barrier method designed for stress-constrained mass minimization problems with discrete material options. The advantages of the full-space barrier method are twofold. First, in the full-space the stress constraints are provably concave, which facilitates the construction of convex subproblems within the optimization algorithm. Second, by using the full-space, it is no longer necessary to employ stress constraint aggregation techniques to reduce adjoint-gradient evaluation costs. The proposed optimization algorithm uses a Newton method where an approximate linearization of the KKT conditions is solved inexactly at each iteration using a preconditioned Krylov subspace method. Sparse constraints that arise in the discrete material parametrization are treated using a null-space method. Results of the proposed algorithm are demonstrated on a series of three topology and multimaterial optimization problems with selection between isotropic and orthotropic materials, as well as discrete ply-angle selection.     

Formal languages for integer programming modeling of shift scheduling problems

This paper approaches the problem of modeling optimization problems containing substructures involving constraints on sequences of decision variables. Such constraints can be very complex to express with Mixed Integer Programming (MIP). We suggest an approach inspired by global constraints used in Constraint Programming (CP) to exploit formal languages for the modeling of such substructures with MIP. More precisely, we first suggest a way to use automata, as the CP regular constraint does, to express allowed patterns for the values taken by the constrained sequence of variables. Secondly, we present how context-free grammars can contribute to formulate constraints on sequences of variables in a MIP model. Experimental results on both approaches show that they facilitate the modeling, but also give models easier to solve by MIP solvers compared to compact assignment MIP formulations.     

Supervised dimensionality reduction via sequential semidefinite programming

Many dimensionality reduction problems end up with a trace quotient formulation. Since it is difficult to directly solve the trace quotient problem, traditionally the trace quotient cost function is replaced by an approximation such that the generalized eigenvalue decomposition can be applied. In contrast, we directly optimize the trace quotient in this work. It is reformulated as a quasi-linear semidefinite optimization problem, which can be solved globally and efficiently using standard off-the-shelf semidefinite programming solvers. Also this optimization strategy allows one to enforce additional constraints (for example, sparseness constraints) on the projection matrix. We apply this optimization framework to a novel dimensionality reduction algorithm. The performance of the proposed algorithm is demonstrated in experiments by several UCI machine learning benchmark examples, USPS handwritten digits as well as ORL and Yale face data.     

An Efficient Solution to Systems of Multivariate Polynomial Using Expression Trees

In recent years, several quite successful attempts have been made to solve systems of polynomial constraints, using geometric design tools, exploiting the availability of subdivision-based solvers [7], [11], [12], [15]. This broad range of methods includes both binary domain subdivision as well as the projected polyhedron method of Sherbrooke and Patrikalakis [15]. A prime obstacle in using subdivision solvers is their scalability. When the given constraint is represented as a tensor product of all its independent variables, it grows exponentially in size as a function of the number of variables. In this work, we show that for many applications, especially geometric ones, the exponential complexity of the constraints can be reduced to a polynomial by representing the underlying structure of the problem in the form of expression trees that represent the constraints. We demonstrate the applicability and scalability of this representation and compare its performance to that of tensor product constraint representation through several examples.     

Improving Linear Constraint Propagation by Changing Constraint Representation

Propagation based finite domain solvers provide a general mechanism for solving combinatorial problems. Different propagation methods can be used in conjunction by communicating through the domains of shared variables. The flexibility that this entails has been an important factor in the success of propagation based solving for solving hard combinatorial problems. In this paper we investigate how linear integer constraints should be represented in order that propagation can determine strong domain information. We identify two kinds of substitution which can improve propagation solvers, and can never weaken the domain information. This leads us to an alternate approach to propagation based solving where the form of constraints is modified by substitution as computation progresses. We compare and contrast a solver using substitution against an indexical based solver, the current method of choice for implementing propagation based constraint solvers, identifying the relative advantages and disadvantages of the two approaches. In doing so, we investigate a number of choices in propagation solvers and their effects on a suite of benchmarks.     

雅克比矩阵近似更新的三维装配约束求解研究

提出了三维装配约束求解中雅克比矩阵近似更新的方法。该方法通过对 迭代过程中满秩以及行秩秩亏雅克比矩阵进行近似更新,提高了约束求解的效率。首先在非 线性迭代求解过程中添加雅克比矩阵及其逆矩阵近似更新的公式;然后给出使用近似更新公 式需要满足的限制条件;最后通过对奇异点扰动算法的描述介绍迭代求解过程中雅克比矩阵 发生行秩秩亏的处理办法。文中提出的策略与算法已在三维装配约束求解引擎CBABench 中实现,给出的实例表明本文提出的方法效果显著。     

Robust design based on constraint networks

This paper describes a robust design method using constraint networks. As opposed to the traditional statistical robust methodology, the proposed method gives a valid model to analyze parameter uncertainties so as to predict conflicts in concurrent design. The mathematical model, which reflects the requirements of robust design, is given in the paper. A general consistency algorithm is designed using interval arithmetic to refine the intervals. This paper also proves that the consistency algorithm is arc consistent if the constraint network is integrated. The constraint network uses the consistency algorithm to verify the design process early in the process and to assist the designers in determining design variables to reduce the multidisciplinary iterations in concurrent design. The quantitative effect of downstream constraints can be analyzed before determining design parameters and potential conflicts can be predicted. A layout design example shows the validity of the method.     

Efficient nonlinear solvers for Laplace–Beltrami smoothing of three-dimensional unstructured grids

The Laplace–Beltrami system of nonlinear, elliptic, partial differential equations has utility in the generation of computational grids on complex and highly curved geometry. Discretization of this system using the finite-element method accommodates unstructured grids, but generates a large, sparse, ill-conditioned system of nonlinear discrete equations. The use of the Laplace–Beltrami approach, particularly in large-scale applications, has been limited by the scalability and efficiency of solvers. This paper addresses this limitation by developing two nonlinear solvers based on the Jacobian-Free Newton–Krylov (JFNK) methodology. A key feature of these methods is that the Jacobian is not formed explicitly for use by the underlying linear solver. Iterative linear solvers such as the Generalized Minimal RESidual (GMRES) method do not technically require the stand-alone Jacobian; instead its action on a vector is approximated through two nonlinear function evaluations. The preconditioning required by GMRES is also discussed. Two different preconditioners are developed, both of which employ existing Algebraic Multigrid (AMG) methods. Further, the most efficient preconditioner, overall, for the problems considered is based on a Picard linearization. Numerical examples demonstrate that these solvers are significantly faster than a standard Newton–Krylov approach; a speedup factor of approximately 26 was obtained for the Picard preconditioner on the largest grids studied here. In addition, these JFNK solvers exhibit good algorithmic scaling with increasing grid size.     

A classification of methods for distributed system optimization based on formulation structure

Augmented Lagrangian coordination (ALC) is a provably convergent coordination method for multidisciplinary design optimization (MDO) that is able to treat both linking variables and linking functions (i.e. system-wide objectives and constraints). Contrary to quasi-separable problems with only linking variables, the presence of linking functions may hinder the parallel solution of subproblems and the use of the efficient alternating directions method of multipliers. We show that this unfortunate situation is not the case for MDO problems with block-separable linking constraints. We derive a centralized formulation of ALC for block-separable constraints, which does allow parallel solution of subproblems. Similarly, we derive a distributed coordination variant for which subproblems cannot be solved in parallel, but that still enables the use of the alternating direction method of multipliers. The approach can also be used for other existing MDO coordination strategies such that they can include block-separable linking constraints. This work is funded by MicroNed, grant number 10005898.     

Interactive design exploration for constrained meshes

In architectural design, surface shapes are commonly subject to geometric constraints imposed by material, fabrication or assembly. Rationalization algorithms can convert a freeform design into a form feasible for production, but often require design modifications that might not comply with the design intent. In addition, they only offer limited support for exploring alternative feasible shapes, due to the high complexity of the optimization algorithm.We address these shortcomings and present a computational framework for interactive shape exploration of discrete geometric structures in the context of freeform architectural design. Our method is formulated as a mesh optimization subject to shape constraints. Our formulation can enforce soft constraints and hard constraints at the same time, and handles equality constraints and inequality constraints in a unified way. We propose a novel numerical solver that splits the optimization into a sequence of simple subproblems that can be solved efficiently and accurately.Based on this algorithm, we develop a system that allows the user to explore designs satisfying geometric constraints. Our system offers full control over the exploration process, by providing direct access to the specification of the design space. At the same time, the complexity of the underlying optimization is hidden from the user, who communicates with the system through intuitive interfaces.     

Designer’s preferences regarding equality constraints in robust design optimization

Robust design optimization (RDO) problems can generally be formulated by appropriately incorporating uncertainty into the corresponding deterministic optimization problems. Equality constraints in the deterministic problem need to be carefully formulated into the RDO problem because of the strictness associated with their feasibility. In this context, equality constraints have been generally classified into two types: (1) those that must be satisfied regardless of uncertainty, examples include physics-based constraints, such as F = ma, and (2) those that cannot be satisfied because of uncertainty, which are typically designer-imposed, such as dimensional constraints. This paper addresses the notion of preferred degree of satisfaction of deterministic equality constraints under uncertainty. Whether or not a particular equality constraint can be exactly satisfied depends on the statistical nature of the design variables that exist in the constraint and on the designer’s preferences. In this context, this paper puts forth three contributions. First, we develop a comprehensive classification of equality constraints in a way that is mutually exclusive and collectively exhaustive. Second, we present a rank-based matrix approach to interactively classify equality constraints, which systematically incorporates the designer’s preferences into the classification process. Third, we present an approach to incorporate the designer’s intra-constraint and inter-constraint preferences for designer-imposed constraints into the RDO formulation. Intra-constraint preference expresses how closely a designer wishes to satisfy a particular constraint; for example, in terms of its mean and standard deviation. A designer may express inter-constraint preference if satisfaction of a particular designer-imposed constraint is more important than that of another. We present an optimization formulation that incorporates the above discussed constraint preferences, which provides the designer with the means to explore design space possibilities. The formulation entails interesting implications in terms of decision making. Two engineering examples are provided to illustrate the practical usefulness of the developments proposed in this paper.