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2. A detailed example maybe found for instance in Hochkirch et al. (2002).Examples for integral constraints are volume (e.g. the dis-placement volume is between ..., the tank volume equalsto ...) and centroids (e.g. the longitudinal center of buoy-ancy shall shift no more than ...) which constitute directgeometric constraints. Indirect integral constraints oftenoriginate from hydrostatic considerations (e.g. minimummetacentric height as determined via the geometry of thedesign waterline).Differential constraints are fairness (e.g. no hollownessmust be present in a certain hull region) and producibility(e.g. plates should be developable to a given extend) etc.As opposed to positional constraints which usually arestraight forward to evaluate, differential constraints callfor more computational effort and, possibly, even requiresome additional simulations.Figure 1: Feasible hull shape with regard to propellerclearanceFigure 2: Infeasible hull shape due to violation of con-tainer hard pointsAs becomes evident from the examples above many con-straints are of inequality type and can generally be writtenin the formCk(−→ x ) ≤CkMax+τk (1)orCk(−→ x ) ≥CkMin −τk (2)and, hence, converted into the standard formatgk(−→ x ) :=Ck(−→ x )−CkMax≤ τk (3)andgk(−→ x ) :=CkMin −Ck(−→ x ) ≤ τk , (4)respectively. Here τk is a user-specified tolerance. Itequals 0 under strict circumstances but may assume asmall positive value if a constraint violation could be ac-ceptable to investigate the region just outside the feasibledomain.In geometric modeling equality constraints are of less im-portance since they often serve to reduce the number of free variables, see for instance Abt et al. (2001) for moredetails. Therefore, priority is given here to the manage-ment of inequality constraints.Constraint managementConstraint management comprises• handling (set up),• analysis,• monitoring,• assessment.The handling and analysis of constraints is realized insidethe Computer Aided Design tool or outside – the latterfor instance if an elaborate simulation needs to be per-formed such as a Finite Element Method analysis (FEM)or a Computational Fluid Dynamics run (CFD). In orderto monitor and assess a set of constraints it is necessaryto decide on a set of free variables−→ x which might even-tually influence the design – similar to what is done in anautomated optimization. A parametric approach to geo-metric modeling should be followed so as to reduce thedesign’s complexity on the basis of a problem-dependenthigh-level definition, see Birk and Harries (2003).Monitoring and assessmentTrying to circumvent a potential bias, a (quasi-)randomsequence of variants should be generated which can beachieved by applying a Sobol algorithm, see Press et al.(1988). Fig. 3 illustrates the distribution of a Sobol se-quence with 5000 variants. The actual design problemcomprised 14 free variables. A plot in R14 being impossi-ble, projections onto the ˜ x1- ˜ x2-plane and the ˜ x1- ˜ x3-planehave been selected, ˜ x1, ˜ x2 and ˜ x3 being normalized freevariables˜ xi := xi−xminxmax−xmin. (5)Any other combination of free variables would result insimilar figures.The Sobol sequence constitutes a design of experiments(DoE) and brings about the statistical basis for monitor-ing and assessment. (It is also regularly performed earlyin an automated optimization for exploration purposes,see Abt et al. (2003). The key difference is that compu-tational intensive objectives need not be computed at thisstage.)A first assessment of the complexity of the design domainis to compute the ratio between the number of feasibledesigns Nf and the overall number of designs Nt :d := NfNt. (6)An unconstrained problem is then characterized by d = 1while a fully active problem shows d = 0. A weakly con-strained problem may be found if 0.9 < d < 1.0 whileFigure 3: Sobol sequencea constraint dominated problem is given by 0 < d < 0.1.(This is a subjective appreciation which may shift accord-ing to the problem field.)If the domain index is rather small – i.e., just a few de-signs are valid – it is useful to contemplate the relaxationof one or several inequality constraints. The domain in-dex d varies with the limiting values CkMin and CkMaxofall constraints, see Fig. 6 and further discussion below.The variation of d with respect to changes in the specifiedlimiting values therefore provides information on whichconstraints are beneficial to relax so as to gain an in-creased number of feasible designs. Since minor changesare more likely to be acceptable high partial derivativesclose to the original limits are advantageous because evena small constraint relaxation will then already help.When assessing specific constraints it is rather straightforward to distinguish active (gk(−→ x ) ≥ 0) and inactive(gk(−→ x ) < 0) constraints.