Gang Xu

Bernard Mourrain
email: bernard.mourrain@inria.fr

Régis Duvigneau
email: regis.duvigneau@inria.fr

André Galligo
email: andre.galligo@unice.fr

Analysis-suitable volume parameterization of multi-block computational domain in isogeometric applications

Keywords: isogeometric analysis, volume parameterization, multi-block computational domain, heat conduction

Parameterization of computational domain is a key step in isogeometric analysis just as mesh generation is in finite element analysis. In this paper, we study the volume parameterization problem of multi-block computational domain in isogeometric version, i.e, how to generate analysis-suitable parameterization of the multi-block computational domain bounded by B-spline surfaces. Firstly, we show how to find good volume parameterization of single-block computational domain by solving a constraint optimization problem, in which the constraint condition is the injectivity sufficient conditions of B-spline volume parametrization, and the optimization term is the minimization of quadratic energy functions related to the first and second derivatives of B-spline volume parameterization. By using this method, the resulted volume parameterization has no self-intersections, and the isoparametric structure has good uniformity and orthogonality. Then we extend this method to the multi-block case, in which the continuity condition between the neighbor B-spline volume should be added to the constraint term. The effectiveness of the proposed method is illustrated by several examples based on three-dimensional heat conduction problem.

Introduction

Isogeometric analysis (IGA for short) method proposed by Hughes et al. in [START_REF] Hughes | Isogeometric analysis: CAD, finite elements, NURBS, exact geometry, and mesh refinement[END_REF] offers the possibility of bridging the gap between CAD and CAE. The approach uses the same spline representation both for the geometry and for the physical solutions, and thus avoids this costly forth and back transformations. This uniform framework provides more accurate and efficient ways to deal with complex shapes and to approximate the solutions of physical simulation problems. On the other hand, it also rises interesting geometric problems for analysis-suitable modeling tools [START_REF] Scott | Local refinement of analysis-suitable T-splines[END_REF] [START_REF] Nguyen-Thanh | Rotation free isogeometric thin shell analysis using PHT-splines[END_REF] [START_REF] Wang | Adaptive isogeometric analysis using rational PHT-splines[END_REF].

It is well known that mesh generation, which generates a discrete mesh of a computational domain from a given CAD object, is a key and the most time-consuming step in finite element analysis (FEA for short). It consumes about 80% of the overall design and analysis process [START_REF] Bazilevs | Isogeometric analysis using T-Splines[END_REF] in automo-tive, aerospace and ship industry. Parametrization of computational domain in IGA, which corresponds to the mesh generation in FEA, also has some impact on analysis result and efficiency. In particular, arbitrary refinements can be performed on the computational mesh in FEA, but in IGA if we compute with tensor product B-splines, we can only perform refinement operations in each parametric direction by knot insertion or degree elevation. Hence, parameterization of computational domain is also being important for IGA. As it is pointed by Cottrell et al. [START_REF] Cottrell | Isogeometric Analysis: Toward Integration of CAD and FEA[END_REF], one of the most significant challenges towards isogeometric analysis is constructing trivariate spline volume parameterizations from given CAD boundary representation.

From the viewpoint of graphics applications, volume parameterization of 3D models has been studied in [START_REF] Li | Harmonic Volumetric Map-ping for Solid Modeling Applications[END_REF][START_REF] Xia | Parameterization of star shaped volumes using Green's functions[END_REF][START_REF] Xia | Direct product volume parameterization using harmonic fields[END_REF]. As far as we know, there are only a few work on the parametrization of computational domains from the viewpoint of isogeometric applications. T. Martin et al. [START_REF] Martin | Volumetric parameterization and trivariate B-spline fitting using harmonic functions[END_REF] proposed a method to fit a genus-0 triangular mesh by B-spline volume parameterization, based on discrete volumetric harmonic functions; this can be used to build computational domains for 3D IGA problems. E. Cohen et.al. [START_REF] Cohen | Analysis-aware Modeling: Understanding Quality Considerations in Modeling for Isogeometric Analysis[END_REF] proposed the concept of analysis-aware modeling, in which the parameters of CAD models should be selected to facilitate isogeometric analysis. They also demonstrated the influence of parameterization of computational domains by several examples. J.M Escobar et al. proposed a method to construct a trivariate T-spline volume of complex genus-zero solids for isogeometric application [START_REF] Escobara | A new approach to solid modeling with trivariate T-spline based on mesh optimization[END_REF]. However, the proposed method demands a surface triangulation as input data. A variational approach for constructing NURBS parameterization of swept volumes is proposed by M. Aigner et al [START_REF] Aigner | Swept volume parametrization for isogeometric analysis[END_REF]. Given boundary CAD information, approximate implicitization technique is used for parametrization of 2D computational domain in [START_REF] Nguyen | Using approximate implicitization for domain parameterization in isogeometric analysis[END_REF]. In [START_REF] Xu | Optimal analysis-aware parameterization of computational domain in 3D isogeometric analysis[END_REF][28], r-refinement method for generating optimal analysis-aware parameterization of computational domain is proposed. However, it only works for specified analysis problems. A general construction method for analysis-suitable planar B-spline parameterization in two-dimensional isogeometric problem is proposed in [START_REF] Xu | Variational Harmonic Method for Parameterization of Computational Domain in 2D Isogeometric Analysis[END_REF] based on harmonic mapping. In this paper, from the given boundary CAD information, the volume parameterization problem for multi-block computational domain is studied based on the trivariate generalization of the method proposed in [START_REF] Xu | Parameterization of computational domain in isogeometric analysis: methods and comparison[END_REF].

In IGA, the parameterization of a computational domain is determined by control points, knot vectors and the degrees of B-spline objects. For three-dimensional IGA problems, the knot vectors and the degree of the computational domain are determined by the given boundary surfaces. That is, given boundary surfaces, the quality of parameterization of computational domain is determined by the positions of inner control points. Hence, finding a good placement of the inner control points inside the computational domain, is a key issue. A basic requirement of the resulting volume parameterization for IGA is that it doesn't have self-intersections, so that it is an injective map from the parametrization domain to the computational domain. In order to get more accurate simulation results, the isoparametric structure in the computational domain should be as uniform as possible and have orthogonal isoparametric surfaces [START_REF] Brakhage | Application of B-spline techniques to the modeling of airplane wings and numerical grid generation[END_REF]. In this paper, we study the volume parameterization problem of multi-block computational domain in isogeometric version, i.e, how to generate analysis-suitable parameterization of the multi-block computational domain bounded by B-spline surfaces. Our main contributions are: -A constraint optimization framework is proposed to generate a multi-block volume parameterization without self-intersections, and the resulted isoparametric structure has good uniformity and orthogonality; -Some classical results in the field of differential geometry related to parametric surfaces are generalized to the case of trivariate parametric volumes, such as the orthogonal conditions of isoparametric structure and the C 1 conditions between B-spline volumes. -We test the volume parameterization results on heat conduction problem to show the effectiveness of the proposed method.

The remainder of the paper is organized as follows. Section 2 presents some preliminary on B-spline volume parameterization. Section 3 describes the constraint optimization method for single-block volume parameterization of 3D computational domain. Section 4 presents a volume parameterization framework for multi-block computational domain based on the proposed methods in Section 3. Section 5 tests the volume parameterization results on heat conduction problem to show the effectiveness of the proposed method. Finally, we conclude this paper in Section 6.

Preliminary

For a parameterization

σ from P : [a, b] × [c, d] × [e, f ] to Ω ⊂ R 3 , we define the boundary surfaces as the image of {a}×[c, d]×[e, f ], {b}×[c, d]×[e, f ],{c}×[a, b]×[e, f],{d}× [a, b] × [e, f ], {e} × [a, b] × [c, d], {f } × [a, b] × [c, d] by σ.
We say that σ defines a regular boundary if these surfaces do not intersect pairwise, except at their boundary curves and if they have no self-intersection points.

We consider the following trivariate B-spline parameterization

σ : (ξ, η, ζ) ∈ P : = [a, b] × [c, d] × [e, f ] → σ(ξ, η, ζ) : = 0≤i≤l 0≤j≤m 0≤k≤n c i,j,k N p i (ξ)N q j (η)N r k (ζ),
where c i,j,k ∈ R 3 are the control points, N p i (ξ), N q j (η) and N r k (ζ) are B-spline functions of degree p, q and r for a given knot vector on [a, b], [c, d] and [e, f ] . Note that in this paper we use knot vectors with multiple end knots to forces the volumes to interpolate the corner control points.

The derivative of σ(ξ, η, ζ) with respect to ξ can be expressed in terms of the differences ∆ 1 i,j,k

:= (∆ 1,x i,j,k , ∆ 1,y i,j,k , ∆ 1,z i,j,k ) = c i+1,j,k -c i,j,k : ∂ ξ σ = 0≤i≤l-1 0≤j≤m 0≤k≤n ω 1 i,j,k ∆ 1 i,j,k N p-1 i (ξ)N q j (η)N r k (ζ), (1)
where N p-1 i (ξ) is the B-spline function with one degree less in u, ω 1 i,j,k is a positive factor. Similarly, the derivative of σ(ξ, η, ζ) with respect to η and ζ can be expressed as follows

∂ η σ = 0≤i≤l 0≤j≤m-1 0≤k≤n ω 2 i,j,k ∆ 2 i,j,k N p i (ξ)N q-1 j (η)N r k (ζ), (2)
∂ ζ σ = 0≤i≤l 0≤j≤m 0≤k≤n-1 ω 3 i,j,k ∆ 3 i,j,k N p i (ξ)N q j (η)N r-1 k (ζ), (3)
where

∆ 2 i,j,k = (∆ 2,x i,j,k , ∆ 2,y i,j,k , ∆ 2,z i,j,k ) = c i,j+1,k -c i,j,k , ∆ 3
i,j,k = (∆ 3,x i,j,k , ∆ 3,y i,j,k , ∆ 3,z i,j,k ) = c i,j,k+1c i,j,k , and ω 2 i,j,k , ω 3 i,j,k are positive factors.

Non-linear sufficient condition based on Jacobian computation

From [START_REF] Xu | Parameterization of computational domain in isogeometric analysis: methods and comparison[END_REF], if the Jacobian determinant J (σ(ξ, η, ζ)) of trivariate B-spline parameterization satisfies J (σ(ξ, η, ζ)) > 0, then σ(ξ, η, ζ) has no self-intersections.

From (1), ( 2), (3) and the product properties of Bsplines [START_REF] Morken | Some identities for products and degree raising of splines[END_REF] , the Jacobian determinant J (σ(ξ, η, ζ)) of Bspline surface can be computed as follows:

J (σ(ξ, η, ζ)) = σ x ξ σ x η σ x ζ σ y ξ σ y η σ y ζ σ z ξ σ z η σ z ζ = 0≤i≤l-1 0≤j≤m 0≤k≤n 0≤i ′ ≤n 0≤j ′ ≤m-1 0≤k ′ ≤n 0≤i ′′ ≤n 0≤j ′′ ≤m 0≤k ′′ ≤n-1 N p-1 i (ξ)N q j (η)N r k (ζ) N p i ′ (ξ)N q-1 j ′ (η)N r k ′ (ζ) N p i ′′ (ξ)N q j ′′ (η)N r-1 k ′′ (ζ) ω 1 i,j,k ω 2 i ′ ,j ′ ,k ′ ω 3 i ′′ ,j ′′ ,k ′′ ∆ 1,x i,j,k ∆ 2,x i,j,k ∆ 3,x i,j,k ∆ 1,y i ′ ,j ′ ,k ′ ∆ 2,y i ′ ,j ′ ,k ′ ∆ 3,y i ′ ,j ′ ,k ′ ∆ 1,z i ′′ ,j ′′ ,k ′′ ∆ 2,z i ′′ ,j ′′ ,k ′′ ∆ 3,z i ′′ ,j ′′ ,k ′′ = 3l-1 i=0 3m-1 j=0 3n-1 k=0 G ijk N 3p-1 i (ξ)N 3q-1 i (η)N 3r-1 k (ζ) (4)
Hence, the Jacobian of trivariate B-spline parameterization can be represented in the form of trivariate B-spline volume with higher degrees. From the convex hull property of B-splines [START_REF] Farin | Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide[END_REF], we have the following theorem.

Theorem 2.1 If G ijk > 0 in (4), then J(σ(ξ, η, ζ)) > 0, that is, σ(ξ, η, ζ) has no self-intersections.
This is a non-linear sufficient condition with respect to the inner control points. We will use it as constraint term in the constraint optimization method for volume generation in Section 3.

Linear constraint for injectivity.

In order to be self-contained, the linear sufficient condition proposed in [START_REF] Xu | Optimal analysis-aware parameterization of computational domain in 3D isogeometric analysis[END_REF] for injectivity of a B-spline volume parameterization is presented in this subsection.

We denote by C i the convex cone of R i generated by

the half rays R + • ∆ i i,j,k ∆ i i,j,k , i = 1, 2, 3.
If this cone is generated by two opposite vectors, which are on a straight line, we define C i (c) as any half-plane. We say that two cones

C 1 , C 2 are transverse if R • C 1 and R • C 2 intersect only at {0}. We say that three cones C 1 , C 2 , C 3 are cotransverse if -0 is a vertex of the convex hull of C 1 , C 2 , C 3 ;
-the convex hull of the cones C i , C j and the cone C k are transverse for all {i, j, k} = {1, 2, 3}. Given six boundary surfaces described by the controls points c i,0,k , c i,l2,k , c 0,j,k , c l1,j,k , c i,j,0 , c i,j,l3 , with 0 (resp. C 0 2 , C 0 3 ) as the cone generated by the vectors ∆ 1 i,j,0 (c),

≤ i ≤ l 1 , 0 ≤ j ≤ l 2 , 0 ≤ k ≤ l 3 , we define the boundary cone C 0 1 O H 1 O 1 O 2 O 3 T 12 M 12 T 21 T 23 M 23 T 32 T 31 M 13 T 13 H 3 H 2
∆ 1 i,j,l3 (c) for 0 ≤ i ≤ l 1 -1, 0 ≤ j ≤ l 2 (resp. ∆ 2 0,j,k (c), ∆ 2 l2,j,k (c) for 0 ≤ j ≤ l 2 -1, 0 ≤ k ≤ l 3 , ∆ 3 i,0,k (c), ∆ 3 i,l2,k (c) for 0 ≤ i ≤ l 1 , 0 ≤ k ≤ l 3 -1)
. We assume that these boundary surfaces form a regular boundary and that the three boundary cones Based on above the construction, the linear constraint conditions of inner control points for injective trivariate Bspline parameterization can be presented as follows,

C 0 1 , C 0 2 , C 0 3 are cotransverse. Then we can find -a plane H 0 such that ∀u ∈ C 0, * i (i = 1, 2, 3), H 0 (u) > 0, and -a plane H k such that ∀u ∈ C 0, * i ∪ C 0, * j , H k (u) > 0 and ∀u ∈ C 0, * k , H k (u) < 0 for {i, j, k} = {1,
                           H 0 (c i+1,j,k -c i,j,k ) > 0, H 1 (c i+1,j,k -c i,j,k ) < 0, H 2 (c i+1,j,k -c i,j,k ) > 0, H 3 (c i+1,j,k -c i,j,k ) > 0, H 0 (c i,j+1,k -c i,j,k ) > 0, H 1 (c i,j+1,k -c i,j,k ) > 0, H 2 (c i,j+1,k -c i,j,k ) < 0, H 3 (c i,j+1,k -c i,j,k ) > 0, H 0 (c i,j,k+1 -c i,j,k ) > 0, H 1 (c i,j,k+1 -c i,j,k ) > 0, H 2 (c i,j,k+1 -c i,j,k ) > 0, H 3 (c i,j,k+1 -c i,j,k ) < 0, ( 5
) where 0 < i < l 1 , 0 ≤ j < l 2 , 0 ≤ k < l 3 .
This set of conditions provides linear constraints for the injectivity of trivariate B-spline volume parameterization. In the following section, it will be used as constraint term in the quadratic programming method for the case when the boundary cones are cotransverse.

Remark 1. Note that the above two sufficient conditions only ensures the injectivity locally. The global injectivity can be guaranteed by the regularity of given boundary surfaces.

Constraint optimization method for volume parametrization of computational domains

In this section, we aim at finding injective volume parameterization of the computational domain with a uniform and orthogonal isoparametric net.

Initial construction of inner control points

In order to solve this constraint optimization problem, an initial construction of inner control points is required. We rely on the discrete Coons method to generate inner control points as initial value from boundary control points, which can be considered as the trivariate generalization of the method presented in [START_REF] Farin | Discrete coons patches[END_REF].

Suppose that given boundary surfaces are Bspline surfaces, the opposite boundary B-spline surfaces have the same degree, number of control points and knot vectors. Given the boundary control points c 0,j,k , c l,j,k , c i,0,k , c i,m,k , c i,j,0 , c i,j,n , (i = 0, . . . , l, j = 0, . . . , m, k = 0, . . . , n ), then the interior control points c i,j,k can be constructed as follows,

c i,j,k = (1 -i/l)c 0,j,k + i/lc l,j,k + (1 -j/m)c i,0,k +j/mc i,m,k + (1 -k/n)c i,j,0 + k/nc i,j,n -[1 -i/l, i/l]   c 0,0,k c 0,m,k c l,0,k c l,m,k     1 -j/m j/m   -[1 -j/m, j/m]   c i,0,0 c i,0,n c i,m,0 c i,m,n     1 -k/n k/n   -[1 -k/n, k/n]   c 0,j,0 c l,j,0 c 0,j,n c l,j,n     1 -i/l i/l   +(1 -k/n)   [1 -i/l, i/l]   c 0,0,0 c 0,m,0 c l,0,0 c l,m,0     1 -j/m j/m     +k/n   [1 -i/l, i/l]   c 0,0,n c 0,m,n c l,0,n c l,m,n     1 -j/m j/m    
Then the corresponding B-spline volume has the following form

σ(ξ, η, ζ) = l i=0 m j=0 n k=0 c i,j,k N i (ξ)N j (η)N k (ζ).
where N i (ξ), N j (η) and N k (ζ) are B-spline function with knot vectors given by boundary surfaces. This initial construction of inner control points can be considered as an extension of boolean sum of ruled volumes and trilinear volumes constructed from give boundary surfaces as proposed in [START_REF] Elber | Volumetric boolean sum[END_REF] . The resulting inner control points lie in the convex hull of the boundary control points as the sum of the coefficients equals 1. For some given boundary surfaces, this construction may cause some self-intersections, and lead to an improper volume parameterization for IGA.

Trivariate parametric volume with orthogonal and uniform grid

An internal energy function of the computational domain will be used as an optimization term in a constraint optimization method to construct a computational domain with an uniform and orthogonal isoparametric grid.

In a trivariate B-spline parametrization, it has isoparametric surfaces and isoparametric curves with B-spline form as shown in Fig. 2(a). The partial derivatives at a point inside the trivariate volume are also shown in Fig. 2(b). For the trivariate parametric volume with orthogonal isoparametric surfaces, we have the following proposition. Proposition 3.1 A trivariate parametric volume σ(ξ, η, ζ) has isoparametric grid with orthogonal isoparametric surfaces if and only if it satisfies the following condition

σ ξ • σ η = σ ξ • σ ζ = σ η • σ ζ = 0 (6)
The above proposition indicates that a trivariate parametric volume has orthogonal isoparametric surfaces if and only if each isoparametric surface has orthogonal isoparametric net.

By minimizing the following energy functions, one can achieve isoparametric grid with orthogonal isoparametric surfaces,

|σ ξ • σ η | + |σ ξ • σ ζ | + |σ η • σ ζ |dξdηdζ. (7)
In view of the following inequalities,

|σ ξ • σ η | ≤ σ ξ 2 + σ η 2 2 |σ ξ • σ ζ | ≤ σ ξ 2 + σ ζ 2 2 |σ η • σ ζ | ≤ σ η 2 + σ ζ 2 2
we replace [START_REF] Cohen | Analysis-aware Modeling: Understanding Quality Considerations in Modeling for Isogeometric Analysis[END_REF] with

σ ξ 2 + σ η 2 + σ ζ 2 dξdηdζ. (8)
The energy functions (8) can be seen as the trivariate generalization of stretch energy in [START_REF] Celniker | Deformable curve and surface finite elements for free-form shape design[END_REF]. The strain energy of parametric surface is related to the uniformity of the isoparameteric net as shown in [START_REF] Celniker | Deformable curve and surface finite elements for free-form shape design[END_REF]. Then we can generalize the strain energy in trivariate form as follows:

( σ ξξ 2 + σ ηη 2 + σ ζζ 2 (9) +2 σ ξη 2 +2 σ ξζ 2 +2 σ ηζ 2 )dξdηdζ.
In summary, combining (8) and ( 9), we use the following optimization term as objective function,

min ( σ ξ 2 + σ η 2 + σ ζ 2 ) +ω( σ ξξ 2 + σ ηη 2 + σ ζζ 2 +2 σ ξη 2 +2 σ ξζ 2 +2 σ ηζ 2 )dξdηdζ.
where ω is a positive constant.

Non-linear constraint optimization method for volume parameterization

After introducing the quadratic energy function in subsection 3.2, the following constraint optimization algorithm can be obtained based on the non-linear sufficient condition in subsection 2.1: Input: six boundary B-spline surfaces Output: inner control points and the corresponding Bspline volume parameterization -Construct the initial inner control points as in subsection 3.1; -Construct the constraint condition (4) from boundary B-spline surfaces as in Section 2; -Solve the following constraint optimization problem by using sequential quadratic programming (SQP for short) method

min ( σ ξ 2 + σ η 2 + σ ζ 2 ) +ω( σ ξξ 2 + σ ηη 2 + σ ζζ 2 +2 σ ξη 2 +2 σ ξζ 2 +2 σ ηζ 2 )dξdηdζ. s.t. G ijk > 0
-Generate the corresponding B-spline volume parameterization σ(ξ, η, ζ) as computational domain.

Remark 2. SQP is an iterative procedure which models the nonlinear optimization problem for a given iterate x k by a Quadratic Programming (QP for short) subproblem, solves that QP subproblem, and then uses the solution to construct a new iteratex k+1 . This construction is done in such a way that the sequence x k converges to the minimum of the optimization problem.

Remark 3. SQP optimization method may obtain a local minimum. To achieve the global minimum, initial position of inner control points is an important issue. From the examples we have tested in this paper, the initial construction of inner control points gives a good local minimum as final parameterization.

Quadratic programming method for parametrization of computational domain

If the boundary injectivity cones are transverse, we can propose a quadratic programming method for parameterization of computational domain as follows: Input: six boundary B-spline surfaces Output: inner control points and the corresponding Bspline volume parameterization -Construct the initial inner control points as in subsection 3.1; -Construct the constraints condition (5) from boundary B-spline surfaces as in Section 2; -Solve the following constraint optimization problem by using a quadratic programming method, -Generate the corresponding B-spline volume parameterization σ(ξ, η, ζ) as computational domain.

min ( σ ξ 2 + σ η 2 + σ ζ 2 ) +ω( σ ξξ 2 + σ ηη 2 + σ ζζ 2 +2 σ ξη 2 +2 σ ξζ 2 +2 σ ηζ 2 )dξdηdζ. s.t.                            H 0 (c i+1,j,k -c i,j,k ) > 0, H 1 (c i+1,j,k -c i,j,k ) < 0, H 2 (c i+1,j,k -c i,j,k ) > 0, H 3 (c i+1,j,k -c i,j,k ) > 0, H 0 (c i,j+1,k -c i,j,k ) > 0, H 1 (c i,j+1,k -c i,j,k ) > 0, H 2 (c i,j+1,k -c i,j,k ) < 0, H 3 (c i,j+1,k -c i,j,k ) > 0, H 0 (c i,j,k+1 -c i,j,k ) > 0, H 1 (c i,j,k+1 -c i,j,k ) > 0, H 2 (c i,j,k+1 -c i,j,k ) > 0, H 3 (c i,j,k+1 -c i,j,k ) < 0,
The first method is a non-linear constraint optimization algorithm, it can be used also for more general cases, including the case where the boundary injectivity cones are non-transverse. The second method is more efficient because the constraint conditions are linear and easy to compute. Fig. 3 shows a 3D example, which is drawn partly to illustrate the interior information of the volume. The given boundary B-spline surfaces and curves are shown in Fig. 3 (a) and 3 (b). Fig. 3 (c) presents the initial volume parametrization of computational domain constructed by discrete Coons method. There are some self-intersections on the initial parameterization. Fig. 3 (d) shows the final volume parameterization of computational domain without self-intersections constructed by the non-linear constraint optimization method. To illustrate the quality of the parameterization, the details of the B-spline volume parameterization are presented in Fig. 3 (e) and 3 (f). The optimization result in Figure 3 (f) avoids self-intersection and gives more uniform iso-parametric grid. We use the orthogonality colormap to show the orthogonality of volume parameterization on the isoparametric grids. In this paper, the orthogonality colormap is computed according to the value of cos α+cos β+cos γ

3

, where α, β and γ are angles formed by σ ξ , σ η and σ ζ . From Fig. 3(g) and Fig. 3 (h), optimized volume parameterization in Fig. 3 (d) has better orthogonality than Coons volume parameterization . The mean value of cos α+cos β+cos γ

Volume parameterization of multi-block computational domain

In this section, we propose a volume parameterization framework for multi-block computational domain based on the proposed methods in Section 3.

C 1 condition of trivariate B-spline parametric volumes

High continuity is one of the advantages in isogeometric analysis. For volume parameterization of multi-block computational domain, C 1 continuity between neighboring trivariate B-spline parametric volume is often required. Here we will propose the C 1 continuity conditions under which two B-spline parametric volume are differentiable.

Assume that the blocks are the same degree (p, q, r) and C 0 , that is, they share a common boundary surface. Additionally, define each B-spline block over an arbitrary domain, thus we have

σ 1 (ξ, η, ζ) defined over [ξ 0 , ξ 1 ]×[η 0 , η 1 ]× [ζ 0 , ζ 1 ] and σ 2 (ξ, η, ζ) defined over [ξ 1 , ξ 2 ]×[η 0 , η 1 ]×[ζ 0 , ζ 1 ].
In order for σ 1 (ξ, η, ζ) and σ 2 (ξ, η, ζ) to be C 1 we require

∂ ∂ξ σ 1 (ξ, η, ζ)| ξ=ξ1 = ∂ ∂ξ σ 2 (ξ, η, ζ)| ξ=ξ1 (10)
Drawing from the control point interpretation of the partials in Eq.( 1) and applying the chain rule, we have

0≤j≤m 0≤k≤n ω 1,1 i,j,k ∆ 1,1 i,j,k N q j N r k = 0≤j≤m 0≤k≤n ω 1,2 i,j,k ∆ 1,2 i,j,k N q j N r k . (11)
From the linear independence property of B-spline function, we have

ω 1,1 i,j,k ∆ 1,1 i,j,k = ω 1,2 i,j,k ∆ 1,2 i,j,k , i = 0, ..., l, (12)
where ω 1,1 i,j,k and ω 1,2 i,j,k are positive factors,

∆ 1,1 i,j,k = c 1 i+1,j,k -c 1 i,j,k , ∆ 1,2 i,j,k = c 2 i+1,j,k -c 2 i,j,k , in which c 1 i,j,k and c 2 i,j,k are control points of trivariate B- spline volume σ 1 (ξ, η, ζ) and σ 2 (ξ, η, ζ) respectively.
From Eq.( 12), C 1 B-spline blocks satisfy the criteria that the three control points in each row of along their boundary surfaces are collinear, and the collinear points are positioned in the ratio dictated by the domain as shown in Fig. 4.

Multi-block volume parameterization with C 1 constraints

After introducing the C 1 condition for B-spline parametric volumes, a framework for multi-block volume parameterization can be derived. Input: multi-block computational domain with boundary spline surfaces Output: inner control points and the corresponding

( σ λ ξ 2 + σ λ η 2 + σ λ ζ 2 ) +ω( σ λ ξξ 2 + σ λ ηη 2 + σ λ ζζ 2 +2 σ λ ξη 2 +2 σ λ ξζ 2 +2 σ λ ηζ 2 )dξdηdζ. s.t.    G λ ijk > 0, λ = 0, ..., N ω 1,λ1 i,j,k ∆ 1,λ1 i,j,k = ω 1,λ2 i,j,k ∆ 1,λ2 i,j,k , i = 0, ..., l,
-Generate the corresponding C 1 multi-block B-spline volume parameterization as computational domain. Remark 4. The continuity between input neighboring boundary B-spline surfaces on different block should be at least C 1 . Remark 5. The quadratic programming method proposed in Section 3 can also be extended to the multi-block case in a similar way. Different from the single-block case, the control points to be determined on the common surface between two blocks should simultaneously satisfy the geometric conditions derived from these two blocks.

Examples and comparison

Starting from a trivariate B-spline volume as computational domain, a general framework of an isogeometric solver for 3D heat conduction problem [START_REF] Farin | Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide[END_REF] has been implemented as a plugin in the AXEL1 platform, yielding a B-spline volume as solution field. The proposed multi-block constraint optimization methods are implemented as a part of the isogeometric toolbox of the project EXCITING2 . In this paper, we test the different parameterizations of computational domains for the following heat conduction problem,

∇(κ(x)∇T (x)) = f (x) in Ω T (x) = T 0 (x) on ∂Ω D κ(x) ∂T ∂n (x) = Φ 0 (x) on ∂Ω N , (13)
where x are the Cartesian coordinates, T represents the temperature field and κ the thermal conductivity. Dirichlet and Neumann boundary conditions are applied on ∂Ω D and ∂Ω N respectively, T 0 and Φ 0 being the imposed temperature and thermal flux (n unit vector normal to the boundary). f is a user-defined source function. In this paper, the source function is defined as

f (x, y, z) = - π 2 3 sin( πx 3 ) sin( πy 3 ) sin( πz 3 ). ( 14
)
The boundary condition is specified as T 0 (x ) = 0 and Φ 0 (x ) = 0.

For problems with unknown exact solution T , suppose that T h is the approximation solution obtained by isogeometric method, then the discrete error e = T -T h . Hence, a posteriori error assessment can be obtained by resolving the following problem,

∆e = -f + ∆T h in Ω e = 0 on ∂Ω D (15)
The approximation error e from ( 15) also has a Bspline form. In order to achieve more accurate results for above problem, some h-refinement operation should be performed. Then we can obtain a good approximation of error volume. Though it is much more expensive, we can use it as an error assessment method to show the effectiveness of the proposed construction method of computational domain.

Fig. 5 (a) and 5 (b) presents the corresponding solution field obtained from the volume parameterization in Fig. 3 (c) and Fig. 3 (d) . The corresponding simulation error obtained from [START_REF] Gill | Practical Optimization[END_REF] are illustrated in Fig. 5 (c) and 5 (d) with same scale. The final parameterization obtained by the nonlinear constraint optimization method can achieve better simulation results than the initial Coons parametrization, which illustrates that the optimized volume parameterization is analysis-suitable. In order to show the effectiveness of the analysis-suitable volume parameterization, the global cross sectional view of the computational results and error colormap are illustrated. We can find that the optimized parameterization obtained by the non-linear constraint optimization method with C 1 continuity constraints can achieve better simula-tion results than the initial Coons parameterization. In order to show the C 1 continuity between two blocks, we compute the first derivatives at three sampling points for common boundary surfaces between σ 1 and σ 2 and the common boundary surface between σ 3 and σ 2 , which are presented in Table .1. Table 1 The first derivatives at sampling points on the common boundary surfaces in Fig. 6. Quantitative data of volume parameterization method presented in Fig. 3, Fig. 6 and Fig. 7 are summarized in Table.2.

∂σ

Conclusion

Analysis-suitable volume parameterization of computational domain plays an important role in isogeometric analysis as mesh generation in finite element analysis. In this paper, volume parameterization problem of multiblock computational domain in isogeometric applications is studied. This problem is solved in a constraint optimization framework , in which the constraint condition is the injectivity sufficient conditions of B-spline volume parametrization, and the optimization term is the minimization of quadratic energy functions related to the first and second derivatives of B-spline volume parameterization. The resulted volume parameterization has no selfintersections, and the isoparametric structure has good uniformity and orthogonality. Finally, the continuity condition between the neighbor B-spline volume are added into the constraint term to achieve a multi-block volume parameterization. Several examples are presented to show the effectiveness of the proposed method.

In the future, we will investigate the impact of boundary surfaces on the volume parameterization results. For a given CAD model, how to obtain an effective volume partition is also an interesting topic for isogeometric applications.

Fig. 1 .

 1 Fig.1. Construction of constraint plane.

  2, 3}. Such separating planes can be deduced easily from convex hull computations of finite sets of vectors, which are generating the cones C 0 1 , C 0 2 , C 0 3 or their unions. As shown in Fig.1, suppose that O is the origin, O 1 , O 2 and O 3 are the center of the circular cone C 0 1 (c),C 0 2 (c) and C 0 3 (c), respectively. The great-circle arc connecting O 1 and O 2 intersects the circular cone C 0 1 (c) and C 0 2 (c) at T 12 and T 21 . We can construct a orthogonal plane H 1 of plane OT 12 T 21 through OM 12 , where M 12 is the middle points of T 12 T 21 . Let F 12 be the linear equation defining the plane H 1 . Then the vectors in C 0 1 (c) and C 0 2 (c) satisfy F 12 ≤ 0 and F 12 ≥ 0 respectively. Similarly, we can define F 13 and F 23 as the linear equations defining the plane H 2 and H 3 .

  (a) Isoparametric surfaces and curves in a trivariate B-spline volume (b) The partial derivatives at a point inside the volume

Fig. 2 .

 2 Fig.2. Isoparametric structure and partial derivatives in the trivariate B-spline parameterization.

Fig. 3 .

 3 Fig.3. Volume parameterization example by non-linear constraint optimization method. The colormap in (h) has the same scale as (g).

  (a) C 1 B-spline blocks (b) Isoparametric surfaces and control lattices in C 1 B-spline blocks

Fig. 4 .

 4 Fig.4. C 1 condition of trivariate B-spline parametric volumes multi-block B-spline volume parameterization with C 1 constraints -Construct the initial inner control points for each block σ i as in subsection 3.1; -Construct the constraints condition (4) from boundary spline surfaces for each block σ i as in Section 2; -Solve the following constraint optimization problem to obtain the inner control points

  (a) solution field with respect to Coons parameterization (b) solution field with respect to optimized parameterization (c) error colormap with respect to Coons parameterization (d) error colormap with respect to optimized parameterization

Fig. 5 .

 5 Fig.5. Simulation example of non-linear constraint optimization method. The colormap in (d) has the same scale as (c).

Fig. 6

 6 presents a volume parameterization example with four blocks. The given boundary B-spline surfaces and (a) boundary surfaces (b) boundary curves of blocks (c) optimized volume parameterization (d) solution field with respect to Coons parameterization (e) solution field with respect to optimized parameterization (f) error colormap with respect to Coons parameterization (g) error colormap with respect to optimized parameterization

Fig. 6 .

 6 Fig.6. Volume parameterization example of computational domain with four blocks. The colormap in (g) has the same scale as (f). the block partition are shown in Fig. 6 (a) and 6 (b). Fig. 7 (c) shows the optimized volume parameterization of computational domain without self-intersections constructed by the non-linear constraint optimization method. Fig. 6 (d) and 6 (e) shows the corresponding solution field obtained from Coons parameterization and optimized parameterization. The corresponding simulation error obtained from (15) are shown in Fig. 6 (f) and 6 (g) with same scale.In order to show the effectiveness of the analysis-suitable volume parameterization, the global cross sectional view of the computational results and error colormap are illustrated. We can find that the optimized parameterization obtained by the non-linear constraint optimization method with C 1 continuity constraints can achieve better simula-

Fig. 7

 7 shows a volume parameterization example with five blocks. The given boundary spline surfaces and curves are shown in Fig. 7 (a) and 7 (b). Fig. 7 (c)(d) present the optimized volume parameterization of computational domain without self-intersections by the quadratic programming method. In order to illustrate the effectiveness of the proposed method, the global cross sectional view of the orthogonality colormap is shown in Fig.7 (e).

  (a) boundary surfaces (b) boundary curves (c) optimized volume parameterization (d) interior view of optimized volume parameterization (e) interior view of orthogonality colormap

Fig. 7 .

 7 Fig.7. Volume parameterization example of computational domain with five blocks.

Fig. 3 ,Fig. 3 pFig. 6 pFig. 7 p

 3367 Fig.3 p = q = r = 3 125 [0,0,0,0,1,2,2,2,2] 5 2.52 0.2033 Fig.6 p = q = r = 3 512 [0,0,0,0,1,2,3,3,3,3] 9 4.13 0.2567 Fig.7 p = q = r = 2 320 [0,0,0,1,2,2,2] 8 3.35 0.1356

Table 2

 2 Quantitative data for volume parameterization in

	1	∂σ 2	∂σ 2	∂σ 3
	∂ξ	∂ξ	∂η	∂η
	(0,0.16,1)	(0,0.16,1) (-2.21,1.58,0) (-2.21,1.58,0)
	(0,0,1.33)	(0,0,1.33) (0.73,1.03,0) (0.73,1.03,0)
	(0,0.35,2.98) (0,0.35,2.98) (1.07,3.04,0) (1.07,3.04,0)

in the optimized volume parameterization is also given in Table.2.

http://axel.inria.fr/

http://exciting-project.eu/

Acknowledgements

The authors wish to thank all anonymous referees for their valuable comments and suggestions. The authors are supported by the 7th Framework Program of the European Union, project SCP8-218536 "EXCITING". The first author is partially supported by the National Nature Science Foundation of China (No.61004117), the Zhejiang Provincial Natural Science Foundation of China (Nos. Y1090718), the Defense Industrial Technology Development Program(A3920110002), the Scientific Starting Foundation of Hangzhou Dianzi University(No. KYS055611029) and the Open Project Program of the State Key Lab of CAD&CG (A1105), Zhejiang University.


M. sc. agr Imke Gerjets
email: igerjets@tierzucht.uni-kiel.de

Dr. sc. agr Imke Traulsen

Dr. sc. agr Kerstin Reiners

Prof. Dr. med. vet Nicole Kemper

Comparison of virulence gene profiles of Escherichia coli isolates from sows with Coliform mastitis and healthy sows

Keywords: ETEC, ExPEC, multiplex PCR, swine, virulence factors

Coliform mastitis (CM) is not only a serious economical and animal welfare touching problem in dairy cattle, but also in sows after farrowing. Due to this disease, the essential adequate supply with colostrum for the growth and the health of the piglets is not ensured. Besides other influencing factors, Escherichia (E.) coli is of great importance as a causative agent of this multifactorial disease. In this study, E. coli isolates from milk samples of healthy and CM-affected sows were examined for the presence of virulence genes associated with extraintestinal E. coli strains, enterotoxigenic E. coli and other pathogenic E. coli.

The isolated E. coli harbored mainly virulence genes of extraintestinal E. coli strains (especially fimC, ompA, traT, hra, kpsMTII, iroN). The virulence gene spectrum for both samples from CM-affected and healthy sows did not differ significantly.

Particular virulence gene profiles of E. coli isolates from diseased sows were not detected.

This study provides novel insights into the role of E. coli in association with mastitis in sows since it is the first time E. coli isolates from CM-affected sows" milk were analysed for virulence genes. Because there were no differences in the prevalence of E. coli and their virulence-associated genes between healthy and diseased sows, other causative factors seem to have greater influence on the pathogenesis of porcine CM.

A c c e p t e d M a n u s c r i p t Introduction "Coliform mastitis" (CM) is the main symptom of puerperal disorders occurring in sows after farrowing which are subsumed under the term postpartum dysgalactia syndrome (PPDS or PDS) [START_REF] Gerjets | Coliform mastitis in sows: A review[END_REF][START_REF] Klopfenstein | Diseases of the mammary glands and lactation problems[END_REF]. The etiology of CM is multifactorial with husbandry, management, feeding and hygiene as influencing factors [START_REF] Klopfenstein | Diseases of the mammary glands and lactation problems[END_REF], but mainly bacteria are the causative agents for the inflammation. In bacteriological analyses, especially Escherichia (E.) coli was isolated, but the strains were not further investigated for virulenceassociated genes. Strains of E. coli can be broadly classified into three groups by their location and their characteristic virulence genes: commensal E. coli, intestinal pathogenic E. coli (IPEC) colonizing the intestine, and extraintestinal pathogenic E. coli (ExPEC) that reach extraintestinal niches like the urinary tract [START_REF] Russo | Proposal for a new inclusive designation for extraintestinal pathogenic isolates of Escherichia coli: ExPEC[END_REF]. In swine, especially enterotoxigenic E. coli (ETEC) as a pathotype of IPEC are well described as causal agents for severe diseases like diarrhea in neonatal and weaned piglets [START_REF] Casey | Design and evaluation of a multiplex polymerase chain reaction assay for the simultaneous identification of genes for nine different virulence factors associated with Escherichia coli that cause diarrhea and edema disease in swine[END_REF]. The ExPEC pathotypes are e.g. causative for urinary tract infections (uropathogenic E. coli (UPEC)) or septicaemia in pigs [START_REF] Daigle | Mutations in the f165(1)A and f165(1)E fimbrial genes and regulation of their expression in an Escherichia coli strain causing septicemia in pigs[END_REF][START_REF] Krag | Genotypic and phenotypic characterisation of Escherichia coli strains associated with porcine pyelonephritis[END_REF][START_REF] Shpigel | Mammary pathogenic Escherichia coli[END_REF]. A selection of virulence genes known to be associated with ETEC, ExPEC pathotypes and shiga toxin-producing E. coli (STEC) is listed in Table 1.

A new putative pathotype of ExPEC was proposed by [START_REF] Shpigel | Mammary pathogenic Escherichia coli[END_REF]: mammary pathogenic E. coli, with as specific set of virulence genes, which are associated with mastitis in dairy animals. However, up to now epidemiological studies have not shown a common virulence gene profile for these E. coli so far [START_REF] Kaipainen | Virulence factors of Escherichia coli isolated from bovine clinical mastitis[END_REF][START_REF] Srinivasan | Phenotypic and genotypic antimicrobial resistance patterns of Escherichia coli isolated from dairy cows with mastitis[END_REF][START_REF] Wenz | Escherichia coli isolates' serotypes, genotypes, and virulence genes and clinical coliform mastitis severity[END_REF].

Materials and methods

Animals and study design

The investigation was carried out between April 2008 and August 2010 on five multiplication herds in Germany (A -E), supervised by PIC Germany GmbH Schleswig (Table 2). The farms were of high health status and tested free from enzootic pneumonia, rhinitis, Actinobacillus pleuropneumoniae and dysentery. The number of sows housed in the farms ranged from 700 to 1,800. The sows were in different parities (1-9) and of different lines (Landrace, Large White and crossbreds, partly with Duroc).

They were identified as CM-affected when their rectal temperature was above 39.5°C 24 h post partum [START_REF] Furniss | Measurement of rectal temperature to predict mastitis, metritis and agalactia (MMA) in sows after farrowing[END_REF] and the mammary glands showed symptoms of inflammation. In addition, the appearance and the performance of the piglets were evaluated with regard to their behavior and body condition. Healthy half-or full-sib sows from the same farrowing group that farrowed closest in time served as controls.

The half-sib design was chosen due to further studies on the genetic background via genotyping (Preißler et al., unpublished data). In total, 2,005 milk samples were examined (1,026 milk samples from sows with CM and 979 from healthy sows).

Before gathering a collective sample of several teats, mammary glands were cleaned and disinfected with disinfection swabs containing 70% isopropyl-alcohol. The first streams of milk were discarded whereas the followings were milked on transport swabs with Amies medium (transwab, medical wire & equipment, Corsham, A c c e p t e d M a n u s c r i p t 5 England). The milk samples were stored at 4°C before sending them to the laboratory within 72 hours.

Bacteriological analysis

The swabs were incubated in Caso broth for 24 h at 37°C. With a plastic loop, 10 µL of the enrichment were streaked onto Columbia blood agar and Endo agar (both Oxoid, Cambridge, United Kingdom) and incubated aerobically another 24 h at 37°C.

The grown bacteria were differentiated by their morphology, haemolysis on blood agar and Gram staining. Pure cultures were grown on blood agar after another 24 h incubation at 37°C before biochemical confirmation to species level with the identification system API (bioMérieux, Craponne, France).

Escherichia coli isolates were distinguished due to individual morphology on blood agar and API 20E. All isolates were selected for further investigations.

Desoxyribonucleic acid of the identified E. coli strains was prepared by solving a few colonies in 200 µL distilled water. After boiling for 10 min and centrifugation, 3 µL of the supernatant was taken for PCR analysis. The presence of virulence genes associated with ExPEC strains, ETEC and other pathogenic E. coli was determined by multiplex PCR (mPCR) assays for all E. coli isolates, as described by [START_REF] Ewers | Avian pathogenic, uropathogenic, and newborn meningitiscausing Escherichia coli: How closely related are they?[END_REF] and [START_REF] Casey | Design and evaluation of a multiplex polymerase chain reaction assay for the simultaneous identification of genes for nine different virulence factors associated with Escherichia coli that cause diarrhea and edema disease in swine[END_REF].

In total, 2,403 isolates were tested for the presence of 27 virulence genes for the following virulence factors (Table 1): heat labile toxin (LT), heat stable toxin a and b (STI, STII), Shiga toxin (Stx2e), capsular polysaccharide (neuC), group II capsule antigen (kpsMTII), outer membrane protein (ompA), transfer protein (traT), heme receptor gene (chuA), catecholate siderophore receptor (iroN), iron transport system genes (sitD chr., sitD ep.), haemolysin A (hlyA), invasins (gimB, ibeA), serin protease A c c e p t e d M a n u s c r i p t 6 autotransporter (pic), different adhesins and fimbrial genes (afa/draB, fimC, hra, iha, sfa/foCD, K88, K99, 987P, F41, F18) and pathogenicity-associated island marker (RPai (malX)). Controls for molecular assays were avian pathogenic E. coli (APEC) strain IMT2470, UPEC strains IMT7920 and IMT9267 and ETEC strains IMT204, IMT19, IMT4830 and IMT3838 [START_REF] Casey | Design and evaluation of a multiplex polymerase chain reaction assay for the simultaneous identification of genes for nine different virulence factors associated with Escherichia coli that cause diarrhea and edema disease in swine[END_REF][START_REF] Ewers | Avian pathogenic, uropathogenic, and newborn meningitiscausing Escherichia coli: How closely related are they?[END_REF], kindly provided by the Institute of Microbiology and Epizootics of the Free University Berlin.

Statistical analysis

The statistical analysis was performed using the procedures FREQ and CORR from the Statistical Analysis System (SAS Institute Inc., 2005). Chi square-tests were used to analyse differences in virulence gene frequencies between diseased and healthy sows. Statistical significance was indicated in two levels: P<0.05* and P<0.01**.

Pearson correlation coefficients, calculated to show associations between virulence genes, were presented as heatmaps. Heatmaps representing gene prevalence were generated to allow assessment of the virulence genes regarding occurrence and distribution (R Development Core Team, 2009). Virulence genes in the heatmaps were arranged automatically according to their means. Genes with similar means are ordered close together.

Correlations and heatmaps were performed with only those virulence genes detected in more than 1 % of the analysed E. coli strains.

A c c e p t e d M a n u s c r i p t

Results

Escherichia coli strains

Escherichia coli was found in 70.6 % (n=724) of the milk samples of CM-affected and in 77.9 % (n=762) of the milk samples of non-infected sows. In total, 1,132 E.coli isolates from CM-positive samples and 1,271 isolates from CM-negative samples were identified and further examined by mPCR. The median number of isolates in milk samples of both diseased and healthy sows was one (Figure 1).

Of the 2,403 E. coli isolates, 593 harbored one virulence gene, 983 two, 357 three and 369 four or more virulence genes. In 101 E. coli isolates, no virulence-associated genes were found. The E. coli isolates from CM-positive as well as from negative sows had a median number of two virulence genes.

Comparison of virulence gene profiles

A variety of virulence genes was identified consisting of mainly those associated with ExPEC (98.9 % of E. coli isolates from diseased and 99.0 % of E. coli isolates from healthy sows) (Table 1). The highest prevalence was found for the type 1 fimbriae fimC (in 84.7 % of the isolates of diseased and 82.3 % of the isolates of healthy sows) and for the protectins ompA and traT (in 35.3 % and 52.1 % of the isolates from CM-positive, and 37.6 % and 49.8 % of the isolates from CM-negative sows, respectively). Other genes identified in 9.3 to 14.8 % of the E. coli isolates were hra, kpsMTII and iroN. Almost all of the virulence-associated factors were more often detected in E. coli isolates of CM-affected sows than in isolates of healthy sows, except 987P, neuC, ompA and gimB.

The virulence genes hra, chuA, iroN and kpsMTII occurred significantly more frequently in isolates of diseased animals. The same applied for particular A c c e p t e d M a n u s c r i p t 8 combinations of these genes (Table 3), except for the profiles chuA -iroN, kpsMTII -chuA -iroN and kpsMTIIhra -chuA -iroN. Those combinations were also less prevalent in all E. coli isolates. The greatest difference between diseased and healthy sows was found for the virulence gene profile chuA -hra (2.7 % in E. coli of CMpositive and 0.9 % in E. coli of CM-negative sampled sows, respectively). In total, there were no obvious patterns specific for either diseased or healthy sows.

Correlations between virulence genes

Statistical analysis of associations between all virulence factors of the E. coli isolates is shown in Figure 2. Several similar patterns in the heatmaps were visible for virulence genes of strains from CM-positive and negative sows: the gene hlyA is positively associated with chuA and pic; iroN is positively associated with ompA and sitDepi, respectively. Highest positive correlations existed between the genes iroN and sitDepi for both isolates from diseased and healthy sows. The genes traT and fimC were also highly positive correlated, but only in E. coli isolates of CM-negative sows.

Gene prevalence with regard to different seasons and farms

The gene traT was more often found in E. coli isolates of samples of CM-positive sows in winter whereas STI (heat stable toxin a) was only found in summer (Figure 3). The gene chuA occurred more frequently in E. coli isolates of positive sows in winter and autumn and iroN in summer, autumn and winter as well as kpsMTII was always more prevalent in samples of diseased sows. All virulence genes were found more often in E. coli isolates of diseased sows in all seasons except for ompA and traT which were more prevalent in isolates of healthy sows in spring.

A c c e p t e d M a n u s c r i p t 9 However, the differences in occurrence of the genes were greater between the seasons than between CM-positive and negative sows.

The same held true for the influence of the farms on the occurrence of virulence genes. The gene STI was only found in E. coli isolates sampled from farm A whereas kpsMTII was more prevalent in samples from farm D. The gene traT occurred more often in isolates from diseased sows. The gene prevalence on the farms differed only slightly between CM-infected and healthy sows. Differences regarding the occurrence of the mentioned virulence genes in the seasons and farms were significant (P<0.05).

Discussion

The aim of the study was to analyse and compare virulence genes of E. coli isolates from milk samples of CM-positive and CM-negative sows, because virulence gene profiles of E. coli isolates associated with mastitis has not been described so far [START_REF] Kaipainen | Virulence factors of Escherichia coli isolated from bovine clinical mastitis[END_REF][START_REF] Srinivasan | Phenotypic and genotypic antimicrobial resistance patterns of Escherichia coli isolated from dairy cows with mastitis[END_REF][START_REF] Wenz | Escherichia coli isolates' serotypes, genotypes, and virulence genes and clinical coliform mastitis severity[END_REF]. Escherichia coli is the pathogen most frequently isolated in association with porcine puerperal disorders [START_REF] Armstrong | Microflora associated with agalactia syndrome of sows[END_REF][START_REF] Awad Masalmeh | Bakteriologische Untersuchungen bei an puerperaler Mastitis (MMA-Syndrom) erkrankten Sauen verschiedener Tierbestände Österreichs (Bacteriological studies in sows with pueperal mastitis in different herds in Austria)[END_REF]Bertschinger et al., 1977a;[START_REF] Ross | Bacteriologic study of sow agalactia[END_REF]. It was also isolated in high frequencies in milk samples of diseased sows in this investigation as well as in milk from healthy sows.

The detailed analysis of virulence-associated genes of the E. coli isolates revealed only slight differences between isolates of diseased and healthy sows (P<0.05).

Although there were single genes or gene combinations with a greater linkage to E. coli isolates from milk samples of CM-affected sows, there were no specific virulence gene patterns detectable. Heatmaps were performed to allow a visualization of The E. coli strains were isolated using an enrichment of the milk samples. This qualitative culture procedure was used to promote the growth of the E. coli strains, as described before for faecal samples [START_REF] Hussain | Isolation and characterization of enteroaggregative, enterotoxigenic, diffusely adherent Escherichia coli and Salmonella Worthington from human diarrhoeic faecal samples in Kashmir and Secundarbad, India[END_REF][START_REF] Wu | Distribution of virulence genes related to adhesins and toxins in shiga toxin-producing Escherichia coli strains isolated from healthy cattle and diarrheal patients in Japan[END_REF].

Regarding the actual presence of virulence genes, an influence of enrichment procedures has only been described in detail for STEC [START_REF] Vimont | Growth of shiga-toxin producing Escherichia coli (STEC) and bovine feces background microflora in various enrichment protocols[END_REF]), but has not been proven for incubation in Caso-Broth for the applied duration. However, a possible influence on the quantitative proportion of different strains cannot be excluded though faecal contamination of the samples was minimized by a strict sampling protocol.

Escherichia coli strains causing acute coliform mastitis in dairy cattle originate from the animal"s faecally contaminated environment and infect the udder via the teat canal [START_REF] Eberhart | Coliform mastitis[END_REF]. Experiments by [START_REF] Bertschinger | Lowering of the incidence of puerperal mastitis in the sow by protection of the mammae from contamination[END_REF]Bertschinger et al. (1977b), where the mammary glands of sows were protected against faecal contamination, support the theory of a galactogenous route of infection via the teat duct. Like bovine mastitis, porcine mastitis may also resemble urinary tract infection as the infection may be ascending [START_REF] Kaipainen | Virulence factors of Escherichia coli isolated from bovine clinical mastitis[END_REF]. Among others, causative agents of urinary tract infections (UTI) are UPEC, a pathotype of ExPEC. In contrast to commensal E. coli isolates, UPEC harbor more virulence genes encoding virulent capsule antigens, iron acquisition systems, adhesions and secreted toxins [START_REF] Wiles | Origins and virulence mechanisms of uropathogenic Escherichia coli[END_REF]. The virulence genes iroN and fimC are reported as urovirulence factors [START_REF] Russo | IroN functions as a siderophore receptor and is a urovirulence factor in an extraintestinal pathogenic isolate of Escherichia coli[END_REF][START_REF] Wiles | Origins and virulence mechanisms of uropathogenic Escherichia coli[END_REF] and were also identified in high percentages in our study. In a survey by [START_REF] Won | Profiles of virulenceassociated genes of avian pathogenic Escherichia coli isolates from chickens with colibacillosis[END_REF], the presence of 19 virulence- However, fimC has also been frequently detected in non-pathogenic E. coli and is proposed to be not highly associated with the pathogenesis of APEC-infections [START_REF] Kawano | Genotypic analyses of Escherichia coli isolated from chickens with colibacillosis and apparently healthy chickens in Japan[END_REF]. We also found the fimbrial gene fimC in high prevalence in isolates of healthy sows, confirming this theory.

The traT gene, detected in half of the examined E. coli strains, was found in milk of CM-affected dairy cattle, too. Out of 160 Finnish isolates from cows with mastitis, 37 %, and out of 113 Israeli isolates, 41 % harbored traT [START_REF] Kaipainen | Virulence factors of Escherichia coli isolated from bovine clinical mastitis[END_REF]. [START_REF] Nemeth | Serum resistance and the traT gene in bovine mastitis-causing Escherichia coli[END_REF] identified the gene in 43 % of E. coli strains isolated from the milk of cows with mastitis. In another study by [START_REF] Acik | traT and CNF2 genes of Escherichia coli isolated from milk of healthy cows and sheep[END_REF], milk samples from healthy cows and sheep were analysed and traT was present in 62.3 % of all isolates (62.5 % of the isolates from cows and 60 % of the isolates from sheep).

All in all, a spectrum of virulence genes was present in bovine mastitis strains of E. coli, but those strains do not possess specific virulence factors contributing to clinical disease. Serum resistance was the only virulence property of E. coli consistently associated with isolates of coliform mastitis in dairy cattle [START_REF] Barrow | The virulence characteristics of strains of Escherichia coli isolated from cases of bovine mastitis in England and Wales[END_REF][START_REF] Fang | Mastitis-causing Escherichia coli: Serum sensitivity and susceptibility to selected antibacterials in milk[END_REF]. A relationship between traT and serum resistance, however, could not be confirmed [START_REF] Nemeth | Serum resistance and the traT gene in bovine mastitis-causing Escherichia coli[END_REF][START_REF] Vandekerchove | Virulence-associated traits in avian Escherichia coli: Comparison between isolates from colibacillosisaffected and clinically healthy layer flocks[END_REF].

The results and conclusions concerning the virulence genes related to bovine mastitis are comparable to the findings of our study in sows. Specific sow factors, e.g. the individual disposition of the animal, are probably more important and the host defense status is generally accepted as key factor determining the outcome of the disease [START_REF] Burvenich | Severity of E. coli mastitis is mainly determined by cow factors[END_REF]. Current investigations deal with the genetic In conclusion, a variety of virulence genes was detected among the E. coli isolates for both samples from CM-positive and negative sows. The identified virulence genes belonged mainly to the large group of genes related to ExPEC, but a categorization into the pathotype ExPEC only by virulence gene typing was not possible. Many virulence-associated factors (e.g. for iron-uptake systems, fimbriae and other adhesions) are fitness factors which help the bacteria to adapt to and successfully colonize the host so that the line between virulence and fitness properties of E. coli strains is very thin [START_REF] Dobrindt | Patho-)genomics of Escherichia coli[END_REF].

The results of our study support the hypothesis that any given E. coli strain, even those considered to be non-pathogenic, can cause coliform mastitis in sows, if further adversely environmental, genetic or other influencing factors promoting infection are present.

  our study was to analyse the occurrence of different virulence genes in E. coli isolates associated with Coliform mastitis in sows.

  genes of isolates of different CM-status, seasons and farms.

  avian pathogenic E. coli (APEC), another pathotype of ExPEC, was determined, and approximately 95 % of the APEC isolates possessed fimC.

  s c r i p t background of CM via genotyping of diseased and healthy sows (Preißler et al., unpublished data).

  to thank Jens Wolfmueller and Evelyn Lass for technical assistance and especially all farmers involved for their help with the sample acquisition.

Figure 1 :Figure 2 :

 12 Figure 1: Number of E. coli isolates in milk samples from CM-positive and negative sows

Figure 3 :

 3 Figure 3: Heatmaps representing gene prevalence in E. coli isolates (n=2,403) of different CM-status (neg, pos), seasons (spring, summer, autumn, winter) and farms (A, B, C, D, E). Colours range from light yellow (gene found in 1 -5 % of the isolates) to dark red (gene found in 80 -88 % of the isolates).

  Figure 3

Table 2 : Number of milk samples and E. coli isolates of five different farms

 2

	farm	number of milk samples	number of E. coli isolates
		CM-negative	CM-positive	CM-negative	CM-positive
	A	498	501	600	477
	B	13	15	16	21
	C	276	323	460	481
	D	25	20	32	27
	E	167	167	163	126
	total	979	1,026	1,271	1,132

Table 3 : Prevalence of virulence gene profiles in E. coli isolates from clinically CM-diseased and healthy sows (*P<0.05, **P<0.01)

 3

	Virulence gene profile	prevalence of E. coli isolates (%) with respective	P-value
		gene profile from:
		samples (n = 979) of	samples (n = 1,026) of
		CM-negative sows	CM-positive sows
	kpsMTII*	9.99	13.07	0.0178
	chuA*	4.80	6.71	0.0434
	hra*	11.33	14.84	0.0106
	iroN*	9.28	12.37	0.0148
	kpsMTII, chuA**	0.87	2.21	0.0068
	kpsMTII, hra**	2.44	4.42	0.0073
	kpsMTII, iroN**	1.42	3.09	0.0052
	chuA, hra**	0.94	2.65	0.0014
	chuA, iroN	0.24	0.44	0.3823
	hra, iroN*	1.34	2.65	0.0204
	kpsMTII, chuA, hra*	0.63	1.59	0.0231
	kpsMTII, chuA, iroN	0.08	0.27	0.2634
	kpsMTII, hra, iroN**	0.31	1.33	0.0052
	chuA, hra, iroN*	-	0.35	0.0339
	kpsMTII, chuA, hra, iroN	-	0.27	0.0663

Acknowledgements

This research project was funded by the German Federal Ministry of Education and Research (BMBF) in the research programme ""FUGATO -Functional Genome Analysis in Animal Organisms," project "geMMAstructural and functional analysis of the genetic variation of the MMA-syndrome" (FKZ0315138).

Prof. Dr. Lothar Wieler, Dr. Christa Ewers and Ines Diehl from the Institute of Microbiology and Epizootics of the Free University Berlin are greatly acknowledged for their technical support with the mPCR assays and for providing the control strains.

A c c e p t e d M a n u s c r i p t