atmospheres of polygons and knotted polygons in this paper we define two statistics a and a the positive and negative atmospheres of a lattice polygon of fixed length n these statistics have the property that a a pn 2 pn where pn is the number of polygons of length n counted modulo translations we use the pivot algorithm to sample polygons and to compute the corresponding average atmospheres using these data we directly estimate the growth constants of polygons in two and three dimensions we find that where the error bars are 67 confidence intervals and the second error bar in the three dimensional estimate of is an estimated systematic error we also compute atmospheres of polygons of fixed knot type k sampled by the bfacf algorithm we discuss the implications of our results and show that different knot types have atmospheres which behave dramatically differently at small values of n

sets of efficient points in a normed space abstract sets of efficient points in a normed space with respect to the distances to the points of a given compact set are geometrically characterized hull and closure properties are obtained these results are relevant to geometry of normed spaces and are mostly useful in the context of location theory

continuous location theory under majority rule this paper studies from a theoretical point of view a single facility location problem in a normed space it is assumed that the facility has a finite or an infinite number of users and the location is selected under majority rule where between two feasible locations each user prefers the closer one in general normed space three concepts of solutions condorcet points simpson points and plurality points are geometrically described in a two dimensional space practical methods of construction are given as well as the relationships between condorcet points plurality points and medians in the same framework when the norm is polyhedral or the number of users is finite it is proved that each condorcet point is a solution to the associated fermat weber problem

winding angle distribution for planar random walk polymer ring entangled with an obstacle and all that spitzer edwards prager frisch model revisited using a general green function formulation we re derive both i spitzer and his followers results for the winding angle distribution of the planar brownian motion and ii edwards prager frisch results on the statistical mechanics of a ring polymer entangled with a straight bar in the statistical mechanics part we consider both cases of quenched and annealed topology among new results we compute exactly the expectation value of the surface area of the locus of points such that each of them has linking number n with a given closed random walk trajectory ring polymer we also consider the generalizations of the problem for the finite diameter disc like obstacle and winding within a cavity

knottedness in self avoiding walks in this letter the author explores the knottedness of self avoiding walks he was not able to represent the data as a power law

poland scheraga models and the dna denaturation transition poland scheraga models were introduced to describe the dna denaturation transition we give a rigorous and refined discussion of a family of these models we derive possible scaling functions in the neighborhood of the phase transition point and review common examples we introduce a self avoiding poland scheraga model displaying a first order phase transition in two and three dimensions we also discuss exactly solvable directed examples this complements recent suggestions as to how the poland scheraga class might be extended in order to display a first order transition which is observed experimentally

knot localization in adsorbing polymer rings we study by monte carlo simulations a model of a knotted polymer ring adsorbing onto an impenetrable attractive wall the polymer is described by a self avoiding polygon on the cubic lattice we find that the adsorption transition temperature the crossover exponent ensuremath phi and the metric exponent ensuremath nu are the same as in the model where the topology of the ring is unrestricted by measuring the average length of the knotted portion of the ring we are able to show that adsorbed knots are localized this knot localization transition is triggered by the adsorption transition but is accompanied by a less sharp variation of the exponent related to the degree of localization indeed for a whole interval below the adsorption transition one can not exclude a contiuous variation with temperature of this exponent deep into the adsorbed phase we are able to verify that knot localization is strong and well described in terms of the flat knot model

structural optimization using equivalent static loads at all time intervals a quasi static structural optimization for elastic structures under dynamic loads is presented an equivalent static load esl set is defined as a static load set which generates the same displacement field as that from a dynamic load at a certain time multiple esl sets calculated at all the time intervals are employed to represent the various states of the structure under the dynamic load they can cover all the critical states that might happen at arbitrary times the continuous characteristics of a dynamic load are considered by multiple static load sets the calculated sets of esls are utilized as a multiple loading condition in the optimization process a design cycle is defined as a circulated process between an analysis domain and a design domain the analysis domain gives the loading condition needed in the design domain the design domain gives a new updated design to be verified by the analysis domain in the next design cycle the design cycles are iterated until the design converges structural optimization with dynamic loads is tangible by the proposed method standard example problems are solved to verify the validity of the method

loose flat knots in collapsed polymers we consider single ring polymers which are confined on a plane but maintain a fixed three dimensional knotted topology the equilibrium statistics of such systems is studied on the basis of a model on square lattice in which the configurations are represented by n step polygons with a number of self intersections restricted to the minimum compatible with the topology this allows to define the size s of the flat knots and to study their localization properties due to the presence of both excluded volume and attractive interactions the model undergoes a theta transition accurate monte carlo results show that while in the high temperature swollen regime both prime and composite knot components are localized s n n t with t 0 in the low temperature compact phase they are fully delocalized t 1 right at the theta transition weak localization prevails t 0 44 0 02 part of the results can be interpreted by taking into account a dominance of figure eight shapes for the coarse grained knotted polymer configurations and by applying the scaling theory of polymer networks of fixed topology in particular t 3 7 can be conjectured as an exact exponent characterizing the weak knot localization at the theta point

composite models of polygons a composite polygon is composed of a lattice polygon in the square lattice which contains in its interior an internal structure which may also be a lattice polygon or a lattice tree or a lattice animal or a lattice disc or a collection of these the properties of composite polygons are considered in this manuscript in particular i shall consider the growth constants and generating functions of these models as well as the statistical mechanics of interacting models of composite polygons it is shown that there is an adsorption transition of the internal structure on the containing polygon and a transition which corresponds to the inflation of the containing polygon by the internal structure

detecting knots in self avoiding walks in this comment the authors discuss problems in the method recently proposed by windwer ibid vol 22 p l605 1989 for computing knot probability in self avoiding walks on the simple cubic lattice

the coc algorithm part ii topological geometrical and generalized shape optimization abstract after outlining analytical methods for layout optimization and illustrating them with examples the coc algorithm is applied to the simultaneous optimization of the topology and geometry of trusses with many thousand potential members the numerical results obtained are shown to be in close agreement up to twelve significant digits with analytical results finally the problem of generalized shape optimization finding the best boundary topology and shape is discussed

knot probabilities in random diagrams we consider a natural model of random knotting choose a knot diagram at random from the finite set of diagrams with n crossings we tabulate diagrams with 10 and fewer crossings and classify the diagrams by knot type allowing us to compute exact probabilities for knots in this model as expected most diagrams with 10 and fewer crossings are unknots about 78 of the roughly 1 6 billion 10 crossing diagrams for these crossing numbers the unknot fraction is mostly explained by the prevalence of tree like diagrams which are unknots for any assignment of over under information at crossings the data shows a roughly linear relationship between the log of knot type probability and the log of the frequency rank of the knot type analogous to zipf s law for word frequency the complete tabulation and all knot frequencies are included as supplementary data

the linking number and the writhe of uniform random walks and polygons in confined spaces random walks and polygons are used to model polymers in this paper we consider the extension of the writhe self linking number and linking number to open chains we then study the average writhe self linking and linking number of random walks and polygons over the space of configurations as a function of their length we show that the mean squared linking number the mean squared writhe and the mean squared self linking number of oriented uniform random walks or polygons of length n in a convex confined space are of the form o n2 moreover for a fixed simple closed curve in a convex confined space we prove that the mean absolute value of the linking number between this curve and a uniform random walk or polygon of n edges is of the form our numerical studies confirm those results they also indicate that the mean absolute linking number between any two oriented uniform random walks or polygons of n edges each is of the form o n equilateral random walks and polygons are used to model polymers in conditions we use numerical simulations to investigate how the self linking and linking number of equilateral random walks scale with their length

lattice stick number of knots the minimal number of straight line segments required to construct a polygonal presentation of the knot k in the cubic lattice is called the lattice stick number of the knot k denoted by it is known that if the crossing number of k satisfies and the main result of this paper is to improve this to if furthermore we will show that for and which implies that this lower bound cannot be improved except for knots with higher crossing numbers

simpson points in planar problems with locational constraints the polyhedral gauge case in this paper we address the problem of finding simpson points in planar models with locational constraints when distances are measured by polyhedral gauges making use of the results we have stated in a previous paper we show here the existence of a finite set of points in the plane independent of the weights associated with the users that contains at least a simpson point connection between simpson points as a result of a voting process and weber points as the outcome of a planning process are explored it is shown by means of an example that the existing relations for the unconstrained case are no longer true when locational constraints are imposed in order to reconcile both voting and planning processes a biobjective problem is described for which we construct a finite dominating set

torsion of polygons in the torsion of polygons and self avoiding walks in the cubic lattice is a measure of the self entanglement of these objects we consider several definitions of torsion in polygons and introduce a fugacity conjugate to the torsion in our models we study the thermodynamic behaviour of these models using probabilistic methods and rigorous methods from statistical mechanics in particular we prove that at least one of our models has a non analyticity in its free energy corresponding to a transition between phases with high and low torsion

the linking number in systems with periodic boundary conditions periodic boundary conditions pbc are often used for the simulation of complex physical systems using the gauss linking number we define the periodic linking number as a measure of entanglement for two oriented curves in a system employing pbc in the case of closed chains in pbc the periodic linking number is an integer topological invariant that depends on a finite number of components in the periodic system for open chains the periodic linking number is an infinite series that accounts for all the topological interactions in the periodic system in this paper we give a rigorous proof that the periodic linking number is defined for the infinite system i e that it converges for one two and three pbc models it gives a real number that varies continuously with the configuration and gives a global measure of the geometric complexity of the system of chains similarly for a single oriented chain we define the periodic self linking number and prove that it also is defined for open chains in addition we define the cell periodic linking and self linking numbers giving localizations of the periodic linking numbers these can be used to give good estimates of the periodic linking numbers in infinite systems we also define the local periodic linking number associated to chains in the immediate cell neighborhood of a chain in order to study local linking measures in contrast to the global linking measured by the periodic linking numbers finally we study and compare these measures when applied to a pbc model of polyethylene melts

new pattern theorems for square lattice self avoiding walks and self avoiding polygons a general pattern theorem for weighted self avoiding polygons saps and self avoiding walks saws in is obtained the pattern theorem for saps fits into the general framework of the pattern theorem for lattice clusters introduced by madras 1999 ann comb 3 357 84 note that unlike other pattern theorems proved for saps this pattern theorem does not rely on first establishing a relationship between saps and saws these results are applied to obtain pattern theorems for self interacting saps and self interacting saws

on the topology of a polymer ring first we survey the literature on knots and links in theoretical physics next we report a numerical study in which equilibrium configurations of ring polymers in an infinite space or confined to the interior of a sphere are generated by using a new algorithm the a priori probability for the occurrence of a knot is determined numerically the results are compatible with scaling laws of striking simplicity we also study the mutual entanglement of links comparing the gauss invariant with the alexander polynomial

entanglement complexity of self avoiding walks self avoiding walks on three dimensional lattices are flexible linear objects which can be self entangled the authors discuss several ways to measure entanglement complexity for n step walks and prove that these complexity measures tend to infinity with n for small n they use monte carlo methods to estimate and compare the n dependence of two of these complexity measures

collapsing and adsorbing polygons the adsorption transition in the phase diagram of a self interacting lattice polygon is examined the polygon has a nearest neighbour contact fugacity and the interaction between the polygon and an impenetrable wall is modelled by a visit fugacity which is conjugate to the number of vertices of the polygon incident with the wall the partition function of this model is z c y z d p v cp c v c y v z c where p c v c is the number of polygons with c nearest neighbour contacts v visits to the wall and n edges and counted up to translations parallel to the wall the limiting free energy of this model is f c y z d limn 1 1 logz c y z and it is known to be a non analytic function of y for each z 1 for all z2 0 1

the number of smallest knots on the cubic lattice it has been shown that the smallest knots on the cubic lattice are all trefoils of length 24 in this paper we show that the number of such unrooted knots on the cubic lattice is 3496

statistical topology and knotting of fluctuating filaments abstract the aim of these notes is to provide an introduction to the topic of statistical topology with this name we refer to a combination of ideas and techniques from statistical mechanics and knot theory used to study the entanglement properties of fluctuating filaments some questions that we are going to address are the following i what is a knot and how can we identify it ii which is the probability of finding a random curve that is knotted iii how complex are these knots iv how big are they to try to partially answer these questions we will make use of few paradigmatic problems and try to investigate them by providing some state of the art theoretical and numerical techniques exercises and lists of open problems will be provided too

the dimensions of knotted polygons the authors study the dimensions mean square radius of gyration and mean span of self avoiding polygons on the simple cubic lattice with fixed knot type the approach used is a monte carlo algorithm which is a combination of the bfacf algorithm and the pivot algorithm so that the polygons are studied in the grand canonical ensemble but the autocorrelation time is not too large they show that although the dimensions of polygons are sensitive to knot type the critical exponent v and the leading amplitude are independent of the knot type of the polygon the knot type influences the confluent correction to the scaling term and hence the rate of approach to the limiting behaviour

entropic exponents of lattice polygons with specified knot type ring polymers in three dimensions can be knotted and the dependence of their critical behaviour on knot type is an open question we study this problem for polygons on the simple cubic lattice using a novel grand canonical monte carlo method and present numerical evidence that the entropic exponent depends on the knot type of the polygon we conjecture that the exponent increases by unity for each additional factor in the knot factorization of the polygon

on the mean and variance of the writhe of random polygons we here address two problems concerning the writhe of random polygons first we study the behavior of the mean writhe as a function length second we study the variance of the writhe suppose that we are dealing with a set of random polygons with the same length and knot type which could be the model of some circular dna with the same topological property in general a simple way of detecting chirality of this knot type is to compute the mean writhe of the polygons if the mean writhe is non zero then the knot is chiral how accurate is this method for example if for a specific knot type k the mean writhe decreased to zero as the length of the polygons increased then this method would be limited in the case of long polygons furthermore we conjecture that the sign of the mean writhe is a topological invariant of chiral knots this sign appears to be the same as that of an ideal conformation of the knot we provide numerical evidence to support these claims and we propose a new nomenclature of knots based on the sign of their expected writhes this nomenclature can be of particular interest to applied scientists the second part of our study focuses on the variance of the writhe a problem that has not received much attention in the past in this case we focused on the equilateral random polygons we give numerical as well as analytical evidence to show that the variance of the writhe of equilateral random polygons of length n behaves as a linear function of the length of the equilateral random polygon

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

simpson points in planar problems with locational constraints the round norm case the simpson set is one of the most popular solution concepts in location models with voting in this paper we address the problem of finding such a set for planar models with locational constraints when the metric in use is induced by a round norm after formulating the problem as a mathematical program we first propose a general algorithm that enables the evaluation of the corresponding objective function then we show how to solve the problem by proving the existence of a finite dominating set of locations the best of which can be found by inspection using the evaluation algorithm proposed

bounds for the minimum step number of knots confined to slabs in the simple cubic lattice volume confinement is a key determinant of the topology and geometry of a polymer however the direct relationship between the two is not fully understood for instance recent experimental studies have constructed p4 cosmids i e p4 bacteriophages whose genome sequence and length have been artificially engineered and have shown that upon extraction their dna knot distribution differs from that of wild type bacteriophage p4 in particular it was observed that the complexity of the knots decreases sharply with the length of the packed genome this problem is the motivation of this paper here a polymer is modeled as a self avoiding polygon on the simple cubic lattice and the confining condition is such that the polygon is bounded between two parallel planes i e bounded within a slab we estimate the minimum length required for such a polygon to realize a knot type our numerical simulations show that in order to realize a prime knot with up to ten crossings in a 1 slab i e a slab of height 1 one needs a polygon of length strictly longer than the minimum length needed to realize the same knot when there is no confining condition in the case of the trefoil knot we can in fact establish this result analytically by proving that the minimum length required to tie a trefoil in the 1 slab is 26 which is greater than 24 the known minimum length required to tie a trefoil without a confinement condition additionally we find that in the 1 slab not all geometrical realizations of a given knot type are equivalent under bfacf moves this suggests that in certain confined volumes knowing the topology of a polymer is not enough to describe all its states

upper bounds on the minimum length of cubic lattice knots knots have been considered to be useful models for simulating molecular chains such as dna and proteins one quantity that we are interested in is the molecular knot and the minimum number of monomers necessary to realize a knot in this paper we consider every knot in the cubic lattice in particular the minimum length of a knot indicates the minimum length necessary to construct the knot in the cubic lattice diao introduced this term he used minimum edge number instead and proved that the minimum length of the trefoil knot 31 is 24 also the minimum lengths of the knots 41 and 51 are known to be 30 and 34 respectively in this paper we find a general upper bound of the minimum length of a nontrivial knot k except the trefoil knot in terms of the minimum crossing number c k the upper bound is moreover if k is a non alternating prime knot then the upper bound is our work can be considered a direct consequence of the results obtained by the authors in hong et al 2012 arxiv 1209 0048 furthermore if k is n 1 n torus knot then the upper bound is

entanglement complexity of lattice ribbons we consider a discrete ribbon model for double stranded polymers where the ribbon is constrained to lie in a three dimensional lattice the ribbon can be open or closed and closed ribbons can be orientable or nonorientable we prove some results about the asymptotic behavior of the numbers of ribbons withn plaquettes and a theorem about the frequency of occurrence of certain patterns in these ribbons we use this to derive results about the frequency of knots in closed ribbons the linking of the boundary curves of orientable closed ribbons and the twist and writhe of ribbons we show that the centerline and boundary of a closed ribbon are both almost surely knotted in the infinite n limit for an orientable ribbon the expectation of the absolute value of the linking number of the two boundary curves increases at least as fast as n and similar results hold for the twist and writhe

scaling behavior of random knots using numerical simulations we investigate how overall dimensions of random knots scale with their length we demonstrate that when closed non self avoiding random trajectories are divided into groups consisting of individual knot types then each such group shows the scaling exponent of 0 588 that is typical for self avoiding walks however when all generated knots are grouped together their scaling exponent becomes equal to 0 5 as in non self avoiding random walks we explain here this apparent paradox we introduce the notion of the equilibrium length of individual types of knots and show its correlation with the length of ideal geometric representations of knots we also demonstrate that overall dimensions of random knots with a given chain length follow the same order as dimensions of ideal geometric representations of knots

entanglement complexity of semiflexible lattice polygons we use monte carlo methods to study knotting in polygons on the simple cubic lattice with a stiffness fugacity we investigate how the knot probability depends on stiffness and how the relative frequency of trefoils and figure eight knots changes as the stiffness changes in addition we examine the effect of stiffness on the writhe of the polygons

monte carlo results for projected self avoiding polygons a two dimensional model for knotted polymers we introduce a two dimensional lattice model for the description of knotted polymer rings a polymer configuration is modelled by a closed polygon drawn on the square diagonal lattice with possible crossings describing pairs of strands of polymer passing on top of each other each polygon configuration can be viewed as the two dimensional projection of a particular knot we study numerically the statistics of large polygons with a fixed knot type using a generalization of the bfacf algorithm for self avoiding walks this new algorithm incorporates both the displacement of crossings and the three types of reidemeister transformations preserving the knot topology its ergodicity within a fixed knot type is not proven here rigorously but strong arguments in favour of this ergodicity are given together with a tentative sketch of proof assuming this ergodicity we obtain numerically the following results for the statistics of knotted polygons in the limit of a low crossing fugacity we find a localization along the polygon of all the primary factors forming the knot increasing the crossing fugacity gives rise to a transition from a self avoiding walk to a branched polymer behaviour

reply to detecting knots in self avoiding walks the author replies to a comment by sumners and whittington ibid vol 23 p 1471 1990 on the detection of knots in self avoiding walks on a simple cubic lattice

new perspectives in competitive location theory abstract this paper gives new perspectives in competitive location theory by considering new norms in two dimensional problems and by considering for the first time the competitive location problem on a graph the results in competitive location theory are generalized by exploiting an isomorphism to the literature in voting theory and by developing new results for competitive location problems on a graph

writhe and mutual entanglement combine to give the entanglement length we propose a method to estimate n e the entanglement length that incorporates both local and global topological characteristics of chains in a melt under equilibrium conditions this estimate uses the writhe of the chains the writhe of the primitive paths and the number of kinks in the chains in a melt an advantage of this method is that it works for both linear and ring chains works under all periodic boundary conditions does not require knowing the contour length of the primitive paths and does not rely on a smooth set of data we apply this method to linear finitely extendable nonlinear elastic chains and we observe that our estimates are consistent with those from other studies

efficient topology optimization of large dynamic finite element systems using fatigue the present paper modifies and extends the recently developed equivalent static load method for the optimization of dynamically loaded linear elastic finite element systems with a huge number of degrees of freedom in the equivalent static load method dynamic loads have been transformed into equivalent static loads this leads to an equivalent static response optimization with multiple loading conditions instead of a dynamic optimization problem in the present paper the equivalent static load algorithm is modified and extended by introducing a fatigue analysis in the iterative optimization procedure where damage is used as suitable termination criterion of the iteration as well as for the determination of a single and meaningful equivalent static load that leads to maximal damage in the structure during the evolution process the structure is systematically stiffened by using the solid isotropic microstructure with penalization approach until a user defined damage level is reached three standard examples from literature and an industrial application with a large number of degrees of freedom 600 000 demonstrate the computational efficiency of the proposed method

solutions to shape and topology eigenvalue optimization problems using a homogenization method a solution strategy to find the shape and topology of structures that maximize a natural frequency is presented the methodology is based on a homogenization method and the representation of the shape of the structure as a material property the problem is formulated as a reinforcement problem in which a given structure is reinforced using a prescribed amount of material two dimensional plane elasticity problems are considered examples are presented for illustration

counting closed 2 manifolds in tubes in hypercubic lattices we consider closed 2 manifolds 2 manifolds without boundary embedded in tubes in the hypercubic lattice z d for orientable 2 manifolds with fixed genus we prove that the exponential growth rate is independent of the genus and we use this to prove a pattern theorem for manifolds with fixed genus we prove a similar theorem for the non orientable case for if the genus is not fixed then we prove a pattern theorem and use this to show that 2 manifolds with genus less than any fixed number are exponentially rare so the typical genus increases with the size of the manifold in four and higher dimensions we prove that orientable manifolds are exponentially rare and are dominated by non orientable manifolds in four dimensions all except exponentially few 2 manifolds both orientable and non orientable contain a local knotted ball pair

the writhe of a self avoiding walk the writhe of a self avoiding walk in a three dimensional space is the average over all projections onto a plane of the sum of the signed crossings we compute this number using a monte carlo simulation our results suggest that the average of the absolute value of the writhe of self avoiding walks increases as nalpha where n is the length of the walks and alpha approximately 0 5 the mean crossing number of walks is also computed and found to have a power law dependence on the length of the walks in addition we consider the effects of solvent quality on the writhe and mean crossing number of walks

geometrical properties of the fermat weber problem abstract this paper is devoted to the fermat weber problem with mixed gauges in order to take into account nonsymmetric distances a geometrical description of the set of solutions is obtained then the concept of metric hull is defined general sufficient conditions are given such that the metric hull of existing facilities intersects the set of optimal locations particularly it is shown that an optimal location can be found in the metric hull in dimension two whatever the gauges are and in dimension n with the rectilinear norm the connection between metric hull convex hull and octagonal hull is studied

random linking of lattice polygons we study the linking probability of polygons on the simple cubic lattice in particular we consider two polygons each having n edges confined to a cube of side l and ask for the linking probability as a function of n and l we also consider other situations in which the polygons are restricted to be not too far apart but not necessarily confined to a cube we prove several rigorous results and use monte carlo methods to address some questions which we are unable to answer rigorously an interesting feature is that the linking probability is a function of l nv where v is the exponent characterizing the radius of gyration of a polygon

the writhe of a self avoiding polygon we discuss the writhe of a self avoiding polygon on a lattice as a geometrical measure of its entanglement complexity we prove a rigorous result about the dependence of the absolute value of the writhe on the number n of edges in the polygon and use monte carlo methods to estimate the distribution of the writhe both for all polygons with n edges and for the subset of polygons that are trefoils

the knot probability in lattice polygons the incidence of knots in lattice polygons in the face centred cubic lattice is investigated numerically the authors generate a sample of polygons using a pivot algorithm and detect knotted polygons by calculating the alexander polynomial if p0n phi is the probability that the polygon with n edges is the unknot then it is known that lim supn to infinity p0n phi 1n e varies as 0 1 they find that varies as phi 7 6 or 0 9 10 6 the effect of the solvent quality on p0n phi is considered the data show that the probability of a polygon being knotted increases rapidly as the quality of the solvent deteriorates

an engineer s approach to optimal material distribution and shape finding abstract the fundamental aim of shape optimization involves the optimal distribution of mass in space observing global and local design constraints the practical realization in applied mechanics can be performed by introducing a foam like material with variable density as discussed by bendsoe and kikuchi this paper uses for the evaluation of material properties a simple energy approach the application is extended to three dimensional domains multiple loading cases are also considered further the approach is implemented in a preoptimizer for optimal shape finding