various shape manipulation operations by which vague and discrete particle system models are created. As a last step of the particle-based geometric modeling ...
VAGUE MODELING FOR CONCEPTUAL DESIGN Imre Horváth Department of Design Engineering Delft University of Technology The Netherlands Zoltán Rusák Joris S. M. Vergeest György Kuczogi Department of Design Engineering Delft University of Technology The Netherlands
ABSTRACT The creation and manipulation of conceptual shapes need different modeling techniques from what are available in present CAD systems. Conceptual modeling techniques are supposed to provide more flexible interaction and mimic working with true materials. Therefore, this paper proposes a new computer internal modeling technique that supports the natural ways of exteriorizing shape concepts as well as their sophisticated geometric representation. The vague shape model consists of three aspect models, the particle system, the physical coupling, and the singularity network models. The discrete particle system model is composed from point sets that are generated with highly interactive input devices. The volumetric and physical manipulations can be applied to the particle clouds. A particle system model is a domain-oriented representation of a cluster of shape instances. To make behavioral simulation possible, physical couplings such as springs and dampers are specified among the particles. The discrete particle system representation can also be converted into continuous surface models based on the extraction of the shape singularities and natural surfaces.
KEYWORDS Discrete modeling, vague representation, particle systems, physical coupling, shape singularity.
1. INTRODUCTION AND PROBLEM STATEMENT 1.1. Setting the stage
Among the many possible areas of further development of computer aided design, there are two evident ones. One area is man-computer communication, which is blocked by the currently used input devices that represent old paradigms. There exist two coupled ways for advancement, namely: (a) application of novel highly interactive input (HI2) devices, and (b) using more natural forms of communication in design. Another area for improvement is the implementation of a non-deterministic computerinternal modeling. To advance with this problem, extensive research has been done into truly creative and knowledge-intensive modeling techniques. It has been proved that more effective shape conceptualization environment can be established if the communication between the human experts and the computer aided modeling and simulation systems is more purposeful and expressive. A condition of it is however that the behavior of these systems has to be smarter and reactive. The research into new interfaces put the highly interactive means (e.g., haptic, speech interpretation, gesture scanning, holographic display, and virtual reality) into the focus (Chu, C.-C. P. et al, 1997). An apparent advantage of these technologies is that they can express both procedural and contextual modeling commands in a more efficient way. In this respect, they by far surpass the opportunities provided by the keyboard-mouse paradigm. In addition to the advanced communication, computer-aided conceptualization systems are expected to interpret the human modeling actions, smartly react to them, and adapt to the incomplete and
imprecise specifications. The enhanced communication and representation together lead definitively to a higher level of naturalization, and provide better solution for capturing and handling design concepts, semantics and intents. Present computer-aided modeling tools have been developed to support final representation of shapes, rather than shape conceptualization. They generally dictate a way of thinking and working which comes from the mathematical theories applied, rather than from the way of thinking and doing of designers. This is the reason why they received better acceptance in pragmatic shape design than in aesthetics governed (often-called syntagmatic) shape design. Conventional geometric modeling tools are considered somewhat ineffective and constraining in the early phases of shape design. The known geometric modeling methods are especially inflexible in handling uncertainties and modalities. It comes from the fact that almost all of them originate in the theory of differentiable manifolds. Conventional geometric representation is based on well-defined, low level geometric entities (Gomes, A. and Middleditch, A., 1997). Hence, these models can hardly embed any design intent or uncertainties. Having recognized these limitations, many researchers started to study the issues of semantic modeling with features, incorporating discreteness, and simulation of the physical behavior of the design. For example, a survey on the problem of semantic feature modeling and a solution for the related validity maintenance are addressed in (Bidarra, R., 1999). Probabilistic shape representation with membership functions from the fuzzy set theory has been considered in (Martin, R. R., 1994). The concept of the so-called particle-based modeling has also been introduced. It can be applied in microscopic representation of materials, as opposed to a macroscopic continuum approximation (Muraki, S., 1991). This approach has proved to be attractive in applications such as modeling cloths, waterfalls, terrain, clouds, and wall of fire (Reeves, W. T., 1983). Oriented particle systems have also been introduced to describe surfaces (Szeliski, R. and Tonnesen, D., 1992). A particle system of randomly oriented triangles with specified static
moment is applied in (Hilton, T. L. and Egbert, P. K., 1994). These new modeling approaches are supported by the theories of discreteness, therefore, they are more appropriate for modeling discontinuous objects and phenomena. Another appeal of particle models is that they can be subject to physical effects that pull or repulse the individual particles (Wu, Y. et al., 1995). Recently, attention has also been paid to the singularity oriented geometric modeling of free form shapes (Shinagawa, Y. et al., 1996). A singularity theory based deformation modeling for soft objects is presented in (Kunii, T. L. and Gotoda, H., 1990).
1.2. About the research problem and the solution concept Since shape conceptualization and early design activities are typically accompanied by uncertainties (e.g., incompleteness, modality, ambiguity, vagueness and under determination), the computer-aided conceptual design (CACD) tools of the near future should have the capability of coping with them (Yamaguchi, Y. et al., 1992). At the same time, it is also important to imitate working with true materials, and to investigate the physical behavior of the conceptualized object in the early phases of design. This is why our research is directed to find computer-oriented solutions for these expectations. The main idea is to develop a comprehensive methodology for the computer internal vague modeling of shapes and shape related concepts. A pilot implementation is underway in the Integrated Concept Advancement (ICA) project. The methodology of the proposed vague representation (V-rep) is based on the assumptions that: (i) the shape concepts are expressed by means such as voice control, gestural sweeps, spatial sketching, object scanning, surface feature manipulation, forming virtual clay, and geometric model reengineering, (ii) independent from the actually used input mechanisms, the procedural and contextual specifications can be identified and unambiguously separated, and (iii) the metric content of the contextual specifications can be captured as scattered uncertain points. As the first step of conceptual shape modeling, particles are generated from the output of the highly
interactive input devices. The particles are represented as weakly defined 3D points with quantified uncertainty, which is assigned to them as metric occurrence. Then, the particles are aggregated in particle clouds by specifying the distances among them with density distribution functions. The particle clouds are the subjects of various shape manipulation operations by which vague and discrete particle system models are created. As a last step of the particle-based geometric modeling, shape instances are created as discrete point sets by operations called instancing and solidification. They eliminate the vagueness of the weakly defined points and introduce modality in instance generation. To simulate the materialistic behavior, physical connections are specified between the close neighbor particles as couplings. This way, it becomes possible to describe the movement of individual particles and particle clouds, collision and sticking together of particle clouds, as well as deformation and crack up of individual particle clouds. This functionality allows us to create and manipulate virtual materials. In order to apply the vague modeling as a front end to conventional geometric modeling, the discrete representation has to be converted to some continuous representation (Mallet, J.L., 1992). For this reason, a singularity network representation has also been incorporated which is an intermediate representation between the discrete particles model and the conventional boundary representation. The basic idea of singularity-based modeling is that the shape of an object is described directly in terms of its non-smooth and smooth boundary domains. To achieve it, the singularity-based representation incorporates a singularity network and a set of natural surfaces. The motivation to apply this idea originates in the form-giving practice where specific modeling elements such as characteristic lines and other shape features are generated first. The shape
singularities are constructed and/or extracted based on the metric relationships among the sharply defined points of the solidified dense point set (Horváth, I. and Vergeest, J. S. M., 1998). Then a singularity network is constructed and the natural surfaces are identified (Lai, M.J., 1997). For a mathematical representation the natural surfaces are broken down into smooth, single connected, four-sided surface patches, represented as NURBS surfaces (Hahn, J. M., 1989), (Pfluger, P. R. et al, 1989). The singularity network and the patch boundaries provide the topological entities, which are needed for conversion of the vague representation to conventional models. The further part of the paper details the key concepts and the implementation of V-rep modeling approach.
2. THE PARTICLE SYSTEM MODEL 2.1. Formalization of non-oriented particle entities The key entity of proposed discrete and vague shape modeling is a particle, ? i , which has metric and materialistic properties concurrently. It is derived from the scattered points generated with some input device. Actually, application of all input mechanisms, i.e., voice-based specification, gestural sweeping, spatial sketching, physical and virtual scanning, goes together with fuzziness, that is, the generated or scanned points carry positional inaccuracy. Consider, for example, the scanning of a human body. Most probably the posture of the scanned human, and also the shapes of the scanned body parts, are varying in the course of the scanning procedure. In an analogous way, the unintended motions and the vibration of the human hand during gestural sweeping or sketching, and the impreciseness of scanning result in variations. All these shape variations are detected, which means that the scanned metric data do not return the nominal geometry.
Being aware of the unavoidable or expected variation, we introduced dedicated measures for it. We supposed that the particles have uncertain reference points and a range of occurrence. We specified the geometric points by a reference vector pi and assigned a metric occurrence ? i to it to indicate the vague spatial position of the particle. For reasons originating in manipulation and materialization of shapes, the reference points, pi and pi+1, of two neighboring particles, ? i and ? i+1, are supposed to be different. The metric occurrence, ? i, specifies a spherically bounded finite domain, ? 3, of the Euclidean space ? 3, ? i = ? 3 ? ? 3, and describes the totality of the possible instantaneous positions of a particle, ? i. For physical modeling, the mass and velocity of the particles are considered. Hence, a particle ? i can be described as a quadruplet, ? i = (pi, ? i, ? i, vi). To generate the reference point of a particle, the input data points are redefined with a technique called overturning averaging, which considers the position of a specified number of points scattered in a given neighborhood, as well as the uncertainty with which they represent a shape. We specified the metric occurrence as a scalar value that is the norm of the vector of maximal length, which can be drawn between two extreme ~ i’input points, pi’and pi”. That is, ? i = max?? ( p ~p i-1
?
p ~p i+1
? p
i i+1
i ~ p
~ p i”)??, and ~ p i’? ~ p i”. This is the radius of the spherically bounded finite domain, in which the reference point of a particle, pi, does occur. The value of the radius is also determined by the overturning averaging of the spatial position of the input points. The overturning averaging is applied to points in a given domain only, as shown in Figure 1. The co-ordinates of the reference points pi and the scalar value of the metric occurrence are calculated as follows:
pi , j, l ?
l ? k? 2
?
pi , j, l 5
(1)
l ? k? 2
? ?
1 l ? k? 1 ? p i, j ,l ? p i , j ,l ? 1 2 l ? k? 2
4
In the above equations the symbolic running index k refers to a particle ? , j indicates a containing particle cloud ? , and i denotes a modeled shape, Q. Since the data points are supposed to scatter moderately (i.e., in a noise range), constant number ( = 5) of spatial points is considered at deriving an averaged location. This way the regenerated particle cloud will be smoother than the original point set. This method of calculation is non-unique, since with different groupings of the data points the results are different. Nevertheless, by a consistent application of Equation (1) and (2), on the one hand, the noise in the data can be filtered out, on the other hand, we can quantify the originally existed impreciseness.
i
oscillation of data points generated data points derived reference points
zones of averaging
Note that in order to preserve the absolute discreteness, no intersection is allowed among the metric occurrences. Also note that generally not all particles that can at all be generated from the input data set are used in shape modeling for representational and computational reasons. The chosen number and distribution depends on the morphological character of the shape to be modeled.
2.2. Introducing particle clouds
Figure 1.
(2)
Reference points and metric occurrences of particles
A collection of n particles ? i is a particle cloud, ? , if (i) it has an extent ei,j in ? 3 so as e(? ) ? 0, and e(? ) ? ? 3, (ii) it is absolutely discrete, i.e., card(? ) ? ? , (iii) the metric occurrence of two distinct particles, ? i and ? i+1, has no intersection,
i.e., ? (? i) ? ? (? i+1) = 0, and (iv) the particles ? i populate the extent e(? ) with a density ? i, which is controlled by some density distribution ? i, and (v) physical connection, ? i, exists between a particle ? i and its close-neighbors ? i+l, where ? i ? ? , l ? 0 (Figure 2). The notions brought in above allow us to describe a collection of particles as a quintuplet, ? = (? i, ei, ? i, ? i, ? i). Owing to the introduction of the metric occurrence ? i, the extent ei of ? is never zero. The conditions (iii), (iv) and (v) ensure the true materialized character. In the rest of the paper the term particle cloud will be used to refer to physical substances (objects). That is, a cloud is treated as an entity of one higher level of complexity. Actually, it is obtained by ordering, converting and attributing the set of data points generated by an input device. A cloud represents the shape of a piece of material or a materialized ?'
z
p'i
?k
i
?
? i+1 i+1
?i
p"i
?" i p i+n ? i+n ?
i+n
y x
Figure 2
Interpretation of the metric properties of a particle cloud
object in the form of a discrete set of particles (e.g., it may represent a piece of clay). For this reason, a unique reference frame, ? ? ? , is allocated for each particle cloud. Thus a particle cloud serves as a meaningful (semantic) unit of shape conceptualization. A particle cloud can be either explicitly or implicitly defined. In the former case, the reference position of the included particles is specified. In the latter case, its density, ? i, and/or density distribution, ? i, are specified. The density is the reciprocal value of the distance between the metric occurrences of a particle and its closest neighbor. We also use the notion of density in combination with the measure of solidity of a point cloud. In addition, the metric relationship between the density and the metric occurrence of the particles also plays an important role in the vague modeling. This parameter is referred to as defining intensity, f(? i, ? i). If its density is constant, then a particle cloud is said to be homoeomerous, otherwise it is heteromerous. The change of density of a heteromerous particle cloud is described by one or more density distribution functions, ? i, ? 3 ? ? +. The density functions are real valued, vector-scalar functions that arrange the reference points of a given number of closest neighbor particles. A distribution function generates a particle mesh in the interior of a ? j, provided that the distribution of the particles on the boundary B? is known. Note that the boundary of a discretely represented object mean a fictitious triangular patch system laid over the neighboring extremal point triplets as a closure. Thus, the density and the distribution function jointly specify the metrics of materialization of a particle cloud. With specific distribution functions the particle clouds may be filled up as hollow, shell or other forms. Due to the metric occurrences, which transmit the uncertainty, the depicted shape is always vague. As follows later, we use the distances di between the close-neighbor particles as well as the metric occurrences ? i to influence the intensity (strength) of definition of the shape.
The smaller di and ? i, the more intense the definition of a shape.
2.3. Interpretation of a particle system A combined or disjoint set of discrete particle clouds, ? j, is a particle system, which can represent either an immediate, or an end-result of composing m clouds of particles in a reference frame, ? i. In view of the elementary modeling entities, a particle system is a union of all particles ? i that are either in the interior, or on the boundary of the object, that is, ? ? p i , where
i? ? ,
? i ? i(? )
or ? i ? b(? ),
but
? i ? c(? ). The vague shape can be made visible for the designer by rendering the boundary of the particle system model. A discretely specified object, Qi, is a system of particles, ? , which (i) is created of m particle clouds, ? j ? ? , and j ? m < 8 , (ii) has a connective discrete boundary B so as N B? p Bi , where pBi ? ? , (iii) has a unified i? 1
density distribution, ? i, (iv) has a unique mass Mi, so as M i ?
m
n
? ?
i?1 j?1
?
i, j
, (v) has a consistent
velocity field vi, vi(pi) = v0 for all pi ? Qi., and (vi) has a set of physical properties, ? i.
2.4. Solidification of particle clouds/systems The authors’ fundamental hypothesis is that a finite number of particles can adequately describe the geometric shape of an object. Based on the definitions introduced above, a discretely specified object Qi represents a set of vague geometries, SG,i, which can be derived as instances, ? *i, of the particle system, ? . It is a useful means to describe vague shapes distributing over a given domain, but to get specific instances, we have to reduce the vagueness of the shape representation. This can be achieved by increasing the definition intensity. The process has been called solidification. The metric occurrence and the density of the particles equally influence the definition intensity. The smaller the metric occurrence and
higher the density, the bigger the definition intensity. It means that a sufficiently dense system of strongly defined particles describes a solidly defined object. We say that a particle ? i ? ? i ? ? 3 is geometrically strongly specified if its metric occurrence is smaller than a threshold value, i.e., ? i ? ? 0, where ? 0 is an arbitrary small value, but ? 0 ? 0. Conversely, a particle is said to be weakly specified if ? i ? ? 0. The density has been defined above as concentration of the particles. It is quantified as the measurable maximal distance between any two close-neighbor particles. In practice, there is a threshold density that is needed to adequately express the shape features of a vaguely defined object properly. In a purposeful manner, a particle cloud is considered sufficiently dense if ? i = min{? (? i) - ? (? i+1)} ? ? 0, under the constraint that ? i > 0. With these in mind, we can state that a cloud ? i,j (or a part of it) is solidly defined if it consists of geometrically sharply specified particles ? i and is sufficiently dense, ? i ? ? 0. If the two conditions included in the above definition are not fulfilled, we consider a cloud weakly defined. This kind of cloud represents multiple geometries (more precisely, unlimited number of shape instances), rather than a specific one. The maximal closure ? max defines the possible maximal volume VQ,max and the minimal closure ? min is the possible minimal volume VQ,min that can be derived. We assume that the range between the maximal closure ? max and the minimal closure ? min of the metric occurrences ? i of the boundary particles ? Bi is filled out by vague shapes. Any shape between ? max and ? min can be obtained by a shape instancing function. It selects the actual closure ? inst of the instance shape, and converts the discrete boundary into a solidly defined one.
2.5.
Deriving shape instances
The objective of shape instancing is to generate concrete shape from a vague particle system. For this purpose, we defined an instancing operation, ? , that applies a discrete valued selective function, ? i(? ), to a system of particles, ? , to select those particles, ? Bk, whose reference points, pBk, are on the minimal convex closure (discrete boundary B) of a shape instance ? *i.
introduced specific shape manipulation operations. The operations can be applied to individual clouds, or a group of clouds. Two categories of the operations have been considered so far, depending on c. mix a. form b. stick whether or not mechanical effects are involved or not in modification of the Figure 4. Volumetric shape manipulation operations original shape. The manipulations The selective function ? i(? ), can, for instance, without mechanical effects are called volumetric. be a simple function that selects reference points The manipulations requiring mechanical effects as linear proportion of the extent between the are called physical. maximal closure ? max and the minimal closure In principle, six fundamental operation can be ? min of the metric occurrences ? k of the boundary defined for manipulation of particle clouds by particles ? Bk. considering the involved physical phenomena, Since the derived point set is in general neither i.e., motion, collision, deformation, unification regular enough, nor sufficiently dense, further and separation. The first two relate to the rigid refining is needed. For an automatic control and body behavior, while the latter three are setting of regularity and density, we apply the important for the actual shape manipulation. solidification operation, ? , that (i) reduces the When the shape of the clouds is physically B metric occurrences ? k, of the particles, ? k, on modified, the distance between the closethe discrete boundary, B, of a shape instance, neighbor particles is changed. It means that a new density distribution function has to be ? *i so as to achieve ? k ? ? 0, (ii) generates new generated with a view to the resultant changes. boundary particles, ? B’k, so as to achieve ? k ? ? 0,, (iii) sets the metric occurrence, ? k, of the For volumetric shape manipulation the form, new boundary particles ? B’k, (iv) homogenizes stick and mix operations have been introduced. the density distribution, ? , of the unified system The form operation mimics cutting down a region of particles, ? , and (v) extracts the set of of a piece of material (say, clay) by the movement B reference points p j to describe the shape of an of a shaped knife, without deforming the base object Q. particle cloud. The stick operation brings two After solidification, the object appears as a regular and dense enough discrete point set. Therefore, the geometric character of the shape and its local features can be recognized. For instance, the abrupt local changes can be identified unless they have incomparable small size. Thus, the discretely bounded shape can indeed be an adequate approximation of a continuously bounded solid. The set of points generated by solidification serves as a basis for singularity recognition in the later phase of the vague modeling.
particle clouds into geometric contact without changing their shape. The mix merges two clouds with consideration to the interaction of the internal materialization (densities).
3. MANIPULATION OF THE VAGUE SHAPE
The stick operation (i) identifies the adhering particles on two target clouds, (ii) modifies the actual reference position of the particles to coincide, (iii) combines the clouds into one resultant cloud without deforming their shape, and (iv) generates a new density distribution for the resultant cloud.
3.1. Volumetric operations on particle clouds To modify a particle cloud or to construct an object from a finite number of clouds, we have
The form operation (i) removes certain portions of a target cloud by means of cutting it by a general half-space H, (ii) generates new boundary particles as requested by the solidification, (iii) sets the metric occurrence of the new boundary particles as well as of those that correspond to the half-space, and (iv) generates a new density distribution for the resultant cloud.
The mix operation (i) makes two target clouds and a system of half-spaces overlap (ii) combines the two clouds into one resultant cloud by mixing the common particles, (iii) generates new boundary particles as requested by the solidification, (iv) sets the metric occurrence of the new boundary particles as well as of those that correspond to the half-space, and (v) generates a new density distribution for the resultant cloud.
3.2.
Mechanical operations on particle clouds
The developed mechanical shape manipulation operations are (i) squash, (ii) bend, (iii) distort, (iv) blend, (v) fracture and (vi) twist. The squash operation corresponds to changing the shape of a piece of clay, or any elastic/plastic object, by pressing it with an object having the desired shape. The bend operation applies bending moment or a pair of forces to deform a shape. The distort operation applies various physical effects (e.g., heating, melting, and explosion) by which it changes a shape of a particle cloud. The blend establishes physical connection between two clouds and concurrently modifies their initial shape as implied by the physical effect, the stiffness and the internal materialization of the manipulated shapes. Finally, the twist operation applies a torque or a pair of forces to deform a shape. The implemented functionality of these operations is defined below. The squash operation (i) brings a target cloud and a system of half-spaces into contact, H, (ii) applies a system of forces ? on H, (iii) generates the new shape of the target cloud as deformed by the half-spaces H and forces ? , (iv) generates new boundary particles as requested by the solidification, (v) sets the metric a. squash occurrence of the new boundary particles as well as of those that correspond to the half-spaces H, and (vi) generates a new density distribution for the resultant cloud. The bend operation (i) applies a bending moment M or a pair of forces ? on the target cloud, (ii)
generates the deformed shape of the target cloud, (iii) rearranges the boundary particles as requested by the deformation, (iv) sets the metric occurrence of the new boundary particles, and (vi) generates a new density distribution for the resultant cloud. The distort operation, (i) applies one or more internal physical effect E on a target cloud, (ii) deforms the shape of the target cloud as it is implied by E, and (iii) generates a new density distribution for the resultant cloud. The blend operation, (i) brings two target clouds and a system of half-spaces H into contact, (ii) applies a system of forces ? on H, (iii) deforms the shape of the target clouds as it is implied by the initial shape of the target clouds, the physical connections, the shape of the half-spaces H, and the applied forces ? , (iv) modifies the actual position of the boundary particles in the contact domain, (v) merges the deformed clouds into one resultant cloud (vi) generates new boundary particles as requested by the solidification, (vii) sets the metric occurrence of the new boundary particles as well as of those that correspond to the half-spaces H, and (viii) generates a new density distribution for the resultant cloud. The fracture operation (i) applies one or more external forces F on a target cloud, (ii) deforms the shape of the target cloud as it is implied by F, (iii) defines the location of cracking and/or separation, (iv) generates new boundary particles as requested, (v) sets the metric occurrence of the new (and the existing) boundary particles, and
b. bend
e. fracture
d. blend
Figure 5.
c. distort
f. twist
Physical shape manipulation operations
(vi) generates a new density distribution for the resultant cloud. The twist operation (i) applies torque T or a pair of forces ? on the target cloud, (ii) generates the deformed shape of the target cloud, (iii) rearranges the boundary particles as requested by the deformation, (iv) sets the metric occurrence of the boundary particles, and (vi) generates a new density distribution for the resultant cloud.
4. THE PHYSICAL COUPLING MODEL 4.1. Interpretation of couplings By definition, all particles have physical properties and are capable to respond to certain physical effects. Consequently, both the particles and the clouds formed by them can exhibit observable physical behavior. The shape of the point sets is modified under the physical effects and geometric constraints. Actually, the change in the shape of a particle cloud is determined by the variation of the Euclidean distances, di, between the close-neighbor boundary particles, ? Bi ? ? j. For modeling and simulation of the behavior, we assign connections, called couplings, to all close-neighbor particles. A coupling ? i is a physical connection between a pair of close-neighbor particles, {? i, ? i+1}, where ? i+1 ? ? i and ? i ? ? i+1, to define the variations of the inter-particle energies relative to a rest state. For instance, a coupling, ? i may depict the change of the potential energy, i.e., the elastic ?? (d)ks ( dip, j, k ? dir, j , k ) ? ? ? ? dip, j, k Fi , j , k ? ? Fi , j , k ? 1 ? ? ? dip, j, k ? dip, j, k ? p ? ? d ( ) k ? ? di , j, k d dip, j, k ?? ?? and plastic deformations, but it may also account for other time-dependent physical characteristics (e.g., thermal dilation, surface stress, and damping). Various techniques have already been proposed for continuum representations, for instance, in (Celniker, G. and Gossard, D., 1991), (Terzopoulos, D. and Fleischer, K., 1988) and (Terzopoulos, D. et al., 1987). We have adopted these techniques to our discrete particle representation.
The system of discrete particles and couplings results in a spatial lattice, which resembles a 3D finite element mesh, (Schumaker, L. L., 1993). The specific material properties are imitated using dampings, springs, and frictions among the distinct interacting particles. Depending on the type of coupling, the cloud of particles may behave as rigid, non-rigid, liquid, gas, porous and composite materials. Although the particle cloud is discrete inside, it looks and behaves like a monolith from outside.
4.2. Behavioral simulation in modeling The vague geometric modeling and the physical characterization introduce two aspects of behavioral simulation, i.e., that of the particle clouds and that of the interaction with the environment under some physical effects. The explicit handling of this duality is the added value of our research and development. It opens up new opportunities in, for example, conceptual modeling. We defined the physical effects as fields (e.g., gravity, viscous drag, attraction, repulsion and acceleration) and forces (e.g., contact and dynamic forces). When the forces are considered as unary forces, they act independently on each particle. The so-called n-ary forces are put on some subset of particles. We hardwired the gravity into the particle clouds and we compute it for the modeled object as a concentrated force. We also apply drag effect that makes a particle gradually come to rest by resisting motion in the absence of other influences. As binary forces we employ Hook-type spring force and rigid body collision force. We suppose that a particle system is capable to represent highly non-linear and non-isotropic material behavior. Therefore, we describe the force between two coupled particles with the following equation: (3) where: Fi and Fi+1 are the forces on ? i and ? i+1, respectively, dpi is the distance between ? i and ? i+1, dpi = pi - pi+1, dri is the rest length, ? (di) is the non-linearity function of the spring force magnitude, ks is a spring constant, ? (di) is the non-linearity function of the damping force magnitude, and kd is a damping constant. d ip, j , k is
the time derivative of dpi, is just vi - vi+1, the difference between the two particles’ velocities. In Equation (3) the spring force magnitude is taken to be non-linearly proportional to the difference between the actual distance and the rest distance. The functions ? (di) and ? (di) are dependent on the modeled material. The damping force magnitude is also taken to be non-linearly proportional to the speed difference vi - vi+1. The forces are applied symmetrically on the unrestricted particles. We fix the boundary particles when we need pivots and anchors. Through the consideration of all these properties, we can use the particle clouds as a basis of minimal energy surfacing, or, for instance, finite element-type stress, deformation, thermal state, fracture, and dynamic behavior analyses. In the following part of this paper we focus on the aspect of free form surfacing.
5. SINGULARITY NETWORK MODEL Continuity properties are characteristics for geometric shapes (Rösch, A. et al., 1997). For this reason most of the present CAD/CAE systems follow the continuity theory as far as their concept of geometric modeling is concerned. Continuity can aptly be considered as continuity of r-dimensional manifolds. An obvious advantage of the manifold theory is that it allows for restricting the investigation of continuity to those degrees that are important for engineering, without damaging generality. A manifold is a particular type of non-empty set M of points pi, where M ? ? n. The modeling methodology proposed in this paper for vague representation of geometryrelated design concepts serves as a framework of a vague modeling front end to conventional Brep modelers. The front-end is supposed to be integrated on data level. Therefore, it is necessary to forward the discrete representation of shapes towards a continuous representation. With a proper consideration to the requirements coming from design activities such as shape conceptualization, and early behavioral simulation, the authors decided to follow the principles of the singularity theory, which can be traced back to the early work of Whitney, H., (1955). The main conjecture is that shape singularities are closely related to the topology
spike singularities
Figure 6.
phantom singularities
crest singularities
Sharp and phantom singularities on an object
of objects and allow us to abstract their shapes. In general terms, a shape singularity, ? , is a subset of non-smooth points, pq, of a boundary B (Horváth, I. and Vergeest, J. S. M., 1998). A singularity of a boundary B is in fact an irreducible analytic set. Depending on the nature of discontinuity, different types of singularities can be identified. Figure 6 shows typical examples for the considered shape singularities, which will be more formally defined below. From here on we assume that B is a boundary of some object Q ? ? 3.
5.1. Sharp shape singularities For a specific class of shape singularities, the cardinality of the sub-set ? is also of importance. In the simplest case, there exists only one point p that fails to be an element of a C1 manifold of dimension 2. A non-smooth point p ? B is a zero order shape singularity ? 0 on B if there exists a neighborhood U ? ? 3 of p such that (B ? U) ? p? is a C1 manifold of dimension 2. In a general case, the sub-set ? comprises an infinite number of linearly connected non-smooth points, pq, which establishes a different type of shape singularity. A set ? ? B of non-smooth points, pq, is a first order shape singularity ? 1 on B provided that (i) there exists a neighborhood U ? ? 3 such that (B ? U) - ? is a C1 manifold of dimension 2, (ii) ? is a C1 manifold of dimension 1, and (iii) the non-smooth points, pq, are linearly connected. When (B ? U) - ? results in two disconnected sub-manifolds U’ and U’’ they are individually requested to be C1 manifolds of dimension 2. A zero order shape singularity is called spike, and a first order one is called crest. When singularities are introduced, continuity can be regarded in two ways: (a) continuity of the singularities (as a domain of a manifold), and (b) continuity of the smooth domain(s) of a manifold. It is of course
irrelevant to talk about the smoothness of spikes, but it is applicable for crests. A crest singularity ? 1 is smooth if all points qk ? ? 1 are smooth on ? 1, but are not smooth on M, which contains ? 1. That is, singularity ? 1 is a C1 manifold of dimension 1. The singularity types, ? 0 and ? 1, introduced above are sub-sets of points of G0 discontinuities. This property is called sharpness. We will therefore jointly refer to ? 0 and ? 1 singularities as sharp singularities. There exist however other classes of singularities, which can be defined by considering the dual aspects of continuity, as mentioned above. These second or higher order discontinuities together will be called phantom singularities.
5.2. Higher order shape singularities
? ? . Let also ? ? * > 0 be given, where ? ? * ? ? . A point p ? B is an element of a second order singularity ? 2 if the curvature jump ? ? ? ? ? *, and ? ? * ? 0. Hence, a second order phantom singularity, ? 2, can be a region for which the specific change in the curvature is above some threshold value, ? ? *.
5.3. Typical sharp singularity structures From the point of view of a singularity-based representation, structures formed by sharp singularities, ? 0 and ? 1, are of importance. When they join together in specific forms, we talk about a singularity structure, ? s. It follows from the above definitions, that only an alternately linked sequence of n zero order singularities, ? 0, and n-1 first order singularities, ? 1, can produce a (mathematically correct) singularity structure, ? s. Typical singularity structures can be identified as shown in Figure 7.
Shape features observed as Gn-smooth, n ? 1, can also be defined as shape singularities. For A singularity chain ? C contains arbitrary instance, a third order shape singularity ? 3 exists numbers of ? 0 and ? 1 singularities, has no on B, if the G2-smooth points of a sub-set ? , ? ? branches of singularities and is not cyclic. A B, have a neighborhood U ? ? 3 such that singularity tree, ? T, comprises arbitrary number (B ? U) - ? is a C3 manifold of dimension 2, and of linearly linked chains, ? Ci, and is not cyclic. A ? is a C2 manifold of dimension 2. Since higher order shape singularities can be stripes and not singularity forest, ? F, is composed of arbitrary only patches, when (B ? U) - ? results in two numbers of trees, ? Tj, and linearly linked chains, disconnected sub-manifolds U’ and U’’ they are ? Ci, and is not cyclic. A singularity loop, ? L, individually requested to be C3 manifolds of contains at least one cycle of linearly linked dimension 2. Note that in a general case these chains, ? Ci. The various singularity structures are conditions are necessary, but not sufficient. The either embedded in, or join to the boundary of, a reason is that they may hold for practically every manifold M. Considering the character of ? s, we curve on a Cn manifold. Therefore, a pragmatic approach is needed. Consider for instance the can distinguish embedded, boundary and point p of a C2 manifold of dimension 2. Let H crossing types of arrangements. be a planar half-space. Suppose that there exists The union of the observable singularity a curve M ? H, containing p, which is G2 discontinuous at p, for a given plane H a. embedded spikes b. embedded chains c. embedded tree d. embedded loop intersecting M transversally at p. Let for all such H the maximal h. singularity network g. crossing forest curvature jump e. boundary trees f. boundary loop of M ? H at p be Figure 7. Typical arrangements of sharp singularities
structures is referred to as singularity network, ? ? . In a similar way, singularity structures can be defined for phantom singularities. In principle, the networks of the sharp and phantom singularities can be aggregated to form a mixed singularity network. In order to arrive at a valid topology, a network of sharp singularities, ? ? , is supposed to be mappable onto an open manifold ? of genus g = 2(1-k) by an injective mapping, where k is the number of topological holes H.
5.4. Definition and representation of natural surfaces From the viewpoint of conventional geometric modeling, the boundary B of an object Q is an arrangement of mathematical surfaces. In reality, objects are bounded by physical surfaces, which represent a natural partitioning of the object’s boundary (Figure 8.a). In certain cases, these surfaces cannot be described as one single mathematical (analytic or parametric) surface, especially when they contain embedded singularities. When the concept of singularitybased modeling is applied, the physical surfaces play a more influential role than the mathematical surfaces. Typically, the physical surfaces are bounded by singularity cycles, and may contain embedded, boundary and/or crossing singularity structures. The elimination of all types of singularity structures ? s,i results in natural surfaces, ? i,j. Thus, the shape of an object is the composition of a singularity network, ? ? ,i , and a set, ? , of natural surfaces, ? i.j: S G ,i ? ?
? ,i
? ?
i
? m ? ? ? ? ? i?1
s ,i
? ? ? ?
? ? ? ?
n j? 1
?
i,j
? ? ? ?
(4)
In this manner, the shape is seen on a higher level than pure geometry. A surface is a natural surface, ? i,j, if it: (i) is bounded by a shape singularity loop ? L (or, in special cases, by the natural limits of the embodying physical
surface), (ii) does not contain any singularity structure, including the boundary loop, (iii) is finite, (iv) is smooth, and (v) is not selfintersecting. A natural surface is said to be a weakly defined natural surface, ? w, provided that its bounding singularity loop, ? L, contains at least one phantom singularity ? p. The natural surfaces are components of one and exactly one physical surface and they are supposed to satisfy certain continuity (smoothness) conditions. Intuitively, a natural surface consists of smooth points, that is, for each point a tangent plane T exists. A natural surface is supposed to be a composition of synthetic surfaces. It means they cannot be represented directly as mathematical surfaces. With the conventional terms, a natural surface is a geometrically continuous complex, Ki, of mathematical surface patches. Hence, a complex Ki is a natural surface, ? i,j, if (i) all component surfaces ? i,k are C1 manifolds of dimension 2, and (ii) all connections tj among components ? i,k are of higher, than C1 continuity. These are indicated in Figure 8.b. Distinguishing natural surfaces enables us to handle the surfaces without any consideration to their mathematical representations. The advantage becomes obvious when the surfaces are generated by HI2 devices.
5.5. Recognition of sharp shape singularities In the case of a continuous boundary, shape singularities can be recognized based on specific tangency and curvature characteristics of the concerned domain of the shape. When the data set provided by the highly interactive input devices is used as a basis for extracting shape singularities, more sophisticated techniques are
l
lq
a.
Figure 8.
b.
Natural surfaces of an object
p i
di-1
di
p i-1 ?q
di+1 p i+1
?
r
p 0
?
Figure 9.
Hint generation for sharp singularities
lr p n
needed. The authors apply hint generation, which assumes that the point set is evenly scattered and dense enough everywhere. Hint generation is a computationally efficient way of reasoning out local surface characteristics. Otherwise, it is rather time consuming to evaluate analytical surface representations globally for their continuity properties. Sharp singularities are detected by an intuitive method called three points test. The hint generation algorithm evaluates the discrete neighborhoods U* of individual data points pi. First, the curvature distribution in U* is estimated. We can utilize that point p in an analytic set V in ? n of dimension n - r is smooth if there exists a neighborhood U of p and analytic functions f1, f2, ..., fr whose differentials are linearly independent such that ? r ? f ? ? 0. V? U = U ? ? ? ? i? 0
i
? ?
The hints are generated based on three subsequent points, pi-1, pi, and pi+1 (Figure. 9). The task is to find the points that are in a +/-? sized half-block neighborhood of equidistant xy and xz planes, respectively, defined in ? i. Let us connect the points inside the half-blocks following their relative distances from the origin of ? i in order to form a spatial polyline exz. For all points pi of polyline exz, except for points p0 and pN, lay a line l through points pi-1 and pi+1. Apply a deviation index ? to separate the smooth points psi from the non-smooth points pnsi, based on their relative distances di from the line l. Set the threshold value of the deviation index ? to the value ? 0 that is proportional to the average vectorial distances between the adjacent points. pi
pi-1
pi+1
pi-2 p 0
ri+1 ri
ri+1
ri C i+1
Ci
Figure 10.
Hint generation for phantom singularities
pn
Calculate the actual vectorial distance di. If di ? ? 0, the tested point is smooth, otherwise, it is not. The tested point is most probably in the proximity of a sharp singularity if the angle ? q included by the line lq going through points pi-2 and pi-1 and the line l, furthermore, the angle ? q included by the line l and the line lr going through points pi+1 and pi+2 is significantly greater than the angle ? r included by the lines lq and lr. For the singularities in principal direction y repeat the process by forming blocks parallel to plane yz and using j = 0, ..., m. When for a nonsmooth point p, hints for the existence of a discontinuity can be generated based on more than one polylines going through the point p, it will be classified as a spike singularity ? 0. Otherwise, it is a non-smooth point p belonging to a crest singularity ? 1.
5.6. Recognition of phantom singularities In order to extract non-sharp singularities we have developed the concept of a method, which is termed as four-point test. The four-point test has much in common with the three-point test presented above, but it uses a curvature-implied principle to extract phantom singularities. In fact, it comprises the same steps up to the definition of polylines exz and eyz. Let’s take one of the exz polylines first. By using three subsequent points pi+1, pi and pi-1 construct a circle ci+1 of radius ri+1 (Figure 10). Then, by using the next tree subsequent points pi, pi-1 and pi-2 construct a circle ci of radius ri. The planes of the circles are in general cases not co-planar. Calculate the curvatures with the expression ? ? ? ?r. Let us set the deviation index ? of the test to a value ? 0 that is proportional to the average distances between the adjacent points. The sensitivity index will express the threshold of separation of G2 continuous points of the physical surface from those that are of G1. Apply the deviation index ? = ? 0 to the difference of all pairs of ? , that is ? ? ? ? i+1 ? ? i. If ? ? ? ? 0, the tested point pi is, or is in the very proximity of, a phantom singularity.
6. SOME IMPLEMENTATION ISSUES The above definitions and consideration of alternative computational and application aspects entail a specific data management and database organization for vague modeling of geometrycentered design concepts. Actually, we are implementing a vague model as a compound of three related representations that are used in three different phases of shape conceptualization. The result of the first phase of initial modeling with point clouds results in the particle system representation. An important component of this representation is the cloud structure graph, which describes the connectivity of the incorporated particle clouds. The boundaries and the interior of the clouds of particles are treated differently. The particles inside a particle cloud are specified procedurally by the density distribution function ? i,j. They are computed when it is needed. However, the particles on the boundaries are explicitly stored as entities. For modeling and computational reasons the weakly specified boundary particles and the sharply specified boundary particles are sorted in clusters, which are handled as selfcontained modeling entities. Independently from their nature, all attributes of the particles are defined in the list of particles. To support the second (optional) phase of the vague modeling, the physical couplings representation is generated. The particle system representation is converted quasi-automatically to this representation. The basis for this conversion is formed by the pieces of information and modeling entities provided by the interior and the boundaries of the vague particle model, furthermore, the attributive information obtained from the user interaction. The physical couplings representation, on one hand stores the created spatial lattice that responds to physical effects, and on the other hand, it stores the attributive data of the individual couplings. The singularity-based representation supports the further conversion of the vague particle model into a surfaced model. In this third phase of vague modeling, the types of singularities (? C, ? T, ? F, and ? L) are recognized, extracted and stored in the respective modeling entities. The
lists of singularity chains, trees, forests and loops are merged in a singularity network that corresponds to a ‘topology’ of the singularitybased model. The other fundamental modeling entities are natural surfaces, which decompose to parametric surfaces. The lists of NURBS surfaces and curves carry the geometric information for the synthetic surfaces. The built in automatic conversions contribute to a high level of downstream associativity among the representations.
7. CONCLUSIONS A computational framework is presented for modeling of objects with vague geometries and physical properties. The proposed means support shape conceptualization and early behavioural simulations. The novelty and advantage of this approach lies in the fact that it is intentionally striving after incorporation of underdetermination in the geometric representation. Furthermore, it also incorporates physical properties by which the vague initial model can be used in early simulations and behavioral analysis. The proposed methodology and the CACD system combine an initial portraying by indecisive particle clouds with a singularitybased natural representation of the shapes. The added value of the work presented in this paper comes from two sources: (a) integration of a vague particle system model, a physical coupling model and a singularity based surface model, (b) introduction of intuitive modeling operations that allow us to mimic materialistic manipulation of particle clouds.
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