船体冰载荷的反演确定法.pdf

Determination of Ice Load on Ship Hull by Inverse MethodLIU Ying-hao1,SUOMINEN Mikko2,KUJALA Pentti2,ZHANG Han-xi-zi1,GAO Liang-tian3(1.China Ship Research and Development Academy,Beijing 100101,China;2.Department of AppliedMechanics Marine Technology,Aalto University,Espoo 15300,Finland;3.College of Shipbuildingand Engineering,Harbin Engineering University,Harbin 150001,China)Abstract:The ice load on a ship hull is directly related to the ship safety during a voyage in ice-covered seas.Two inverse methods,influence coefficient matrix method and Tikhonov regularization,areintroduced to determine the ice load on the stern shoulder of a polar supply and research vessel withfull-scaled measurement data.The influence coefficient matrix method is built up experimentally andused to obtain the ice load on frames while Tikhonov regularization is constituted with three discretized approaches and utilized to determine the ice load on the plate field.The results show that bothinverse methods could overcome the ill-conditions of processing data and obtain the ice load on shiphull.Key words:ice load;full-scale measurement;inverse method;load discretizationCLC number:U661.43Document code:Adoi:10.3969/j.issn.1007-7294.2023.06.0100 IntroductionRecently,the increase of marine transportation and activities in Polar Regions has been keptup,since there is a great underground fortune in cold regions and the ice conditions are turning easier.Determination of the ice load acting on a ship structure is important to ensure the safety and stability of the vessel to operate in ice-covered seas.In mechanics,the inverse problems can be distinguished in two categories1-2:solving systemmodel with given inputs and responses and solving inputs with given system model and responses.The ice load determination belongs to the second category of inverse problem.When the measuredresponse and system properties are known,its a process to determine the dynamic load acting onthe structures3.The methods of dynamic load determination have been studied in many areas withthe increasing practical engineering demand in recent years.Doyle4investigated an inverse deconvolution method with wavelet analysis and Fourier analysis to reconstruct the dynamic load of abeam and plate structure.Romppanen et al5presented an inverse sensing method to achieve theline load of press rolls in paper manufacturing.Ikonen et al6determined the ice-induced momenton the propeller of an ice-going vessel based on different inverse methods,such as truncation of sin第27卷第6期船舶力学Vol.27 No.62023年6月Journal of Ship MechanicsJun.2023Article ID：1007-7294（2023）06-0905-14Received date:2022-12-26Foundation item:Supported by the National Key R&D Program of China(2016YFE0202700)Biography:LIU Ying-hao(1988-),female,Ph.D.,E-mail:;SUOMINE Mikko(1985-),male,Ph.D.;KUJALA Pentti(1954-),male,professor;ZHANG Han-xi-zi(1989-),female,MSc.;GAO Liang-tian(1964-),male,professor.gular value decomposition and Tikhonov regularization,etc.However,these achievements couldnot be applied to the research of ice-induced load acting on a ship hull directly due to the complication of structures and sea-ice characteristics.This paper is aimed to determine the ice load on the stern shoulder when a ship navigatesthrough ice-covered areas.Two inverse methods are investigated to determine ice load on differentstructures.The first is influence coefficient matrix method,which determines the frame load basedon the experimental data.The second is Tikhonov regularization,which is used to obtain the hullload.Three different discretizations have been applied on a finite element model to analyze the iceload equation based on Tikhonov regularization.The frame load and hull load are constructed withboth methods and compared based on the full-scale measurement data during a voyage in Antarctica.1 Inverse methodEngineering problems usually are divided into forward and inverse problems based on thecharacteristics of the variables in the problems.The forward problem is to find the output responsesbased on the known system model and inputs while the inverse problem is to obtain the inputs withthe known output responses and system model1.The mathematical form of the inverse method is=Sf(1)where is output response,S is the system model matrix,and f is the input excitation.In this paper,the output response is the strain vector of full-scale measurement on a polar supply and researchvessel,the input excitation is the ice load vector during ship-ice interactions,and the system matrixdepicts the interdependency of the two terms.As is well-known,the inverse load identification problems are ill-posed,which means the solutions are unstable and inaccurate.Therefore,a method has to be found to solve Eq.(1)withoutcausing the ill-condition of the equation.Assuming that a set of linearly separate unit load case vectors constitute the unknown ice load vector f,i.e.f=u(2)where u is the vector of load case factors,and the matrix A consists of the column vectors of the unitload case,=A1A2An(3)Eq.(1)can be written into the form of=S(Au)(4)Now the problem is simplified to determine the load case factors u with the measured strain data.According to the assumptions,the matrix A is known,which is obtained from the model of thestructure.Then the system influence matrix S has to be determined carefully by experimental or numerical methods for avoiding the inverse ill-conditions.If there are m measured strain points and n loading points or areas,three cases below will ariseduring the procedure of determining the system influence matrix:(1)The measured strain points are more than loading points or areas,i.e.m n;906船舶力学第27卷第6期(2)The measured strain points are equal to loading points or areas,i.e.m=n;(3)The measured strain points are smaller than loading points or areas,i.e.m n.For Case(3)the known information is not enough to determine the unknown variables.For Cases(1)and(2)the load case factors are obtained from the following equation:u=-1f(5)For a linear system,the influence matrix can be determined by applying the unit load cases oneach desired points or areas of the structure independently.According to Eq.(1),the element of theinfluence matrix S isSij=fi-1 j(6)where fiis the independent unit load applied on the structure,and jis the strain value measured onthe loading point or area j.Furthermore,the Tikhonov regularization was utilized to overcome the ill-conditions of theprocess to determine ice load,since it has been studied as an effect method to solve the inverseproblems7.Eq.(1)could be transformed into the form with Tikhonov regularization5:p=argminfSf-2+Df2(7)where p is the discretized ice load,S the system matrices,f is the solution vector to minimize the argmin-equation,is the strain vector,is the regularization parameter,and D is the matrix for describing the dependency of the solution elements between each other.The regularization parameter is utilized for balancing the weights of the two terms of Euclideannorm in Eq.(7).The solution might be unstable for too small values of,while too large mightcause an inaccuracy solution.A good starting point for suggested by Doyle8is=Tr(ST S)/Tr(D)(8)where Tr is the trace of the matrix,andthe dependence matrix D is a zeroth-order identity matrix if the solution elements of f are known to have no connection.Otherwise,if the elements of f areknown to be smooth or continuous,thematrix D could be a first-order linearization form8asD=-110.0-11.00-1.(9)The principle of utilizing inversemethods to determine the ice-induced load on a ship hull is shown in Fig.1.2 Ice load determination of a ship structure2.1 Ship dimensions and measurement dataThe ship used for the full-scale measurement is S.A.Agulhas II,which is a PC-5 polar supplyFig.1 Principle of inverse load determination第6期LIU Ying-hao et al:Determination of Ice Load on 907and research vessel.It was built at Rauma shipyard in Finland and completed in April 2012.Themain dimensions of S.A.Agulhas II are presented in Tab.1.Tab.1 Main dimensions of S.A.Agulhas IIParameterLength between perpendicularsBreadth,mouldDraught,designValue121.8 m21.7 m7.65 mParameterDeadweight at design displacementService speedValue5000 t14.0 knThree parts of S.A.Agulhas II were fitted with strain sensors to measure ice load of the shiphull during an Antarctic voyage as illustrated in Fig.2.Fig.2 Strain sensors locations on S.A.Agulhas II9As shown in Fig.2,the three areas under observation from right to left are bow,bow shoulderand stern shoulder respectively.The strain data from the bow and bow shoulder are not considered,since the ice-induced load on the stern shoulder is the focus in this paper.There were 14 strainsensors located at the stern shoulder.Eight sensors numbered from SS16 through SS23 at frameswere utilized to measure shear strains and 6 sensors numbered from SS24 to SS29 at plate field fornormal strains.The experimental voyage sailed from Cape Town,South Africa to Antarctica starting at the endof November,2013 to the middle of February,2014.During the period between 7th December to 1stFebruary,ice was observed and taken for samples.The ice-induced load on the stern shoulder wasexpected to be greater when the vessel turned left or broke out of a channel because all the strainsensors were located on the starboard side.Hence the strain data of two typical ice-stern contactsnamed Ice Contact 1 and Ice Contact 2 were chosen for further analysis,hereinto,Ice Contact 1 wasa starboard ramming and Ice Contact 2 was following a brash channel.The ice conditions of both908船舶力学第27卷第6期ice contacts are described in Tab.2.Tab.2 Ice conditions of ship-ice interactionsStarting timeDuration/sDraught/mBrash ice(%)RammingIce concentration in tenths*Ice thickness in tenthsFloe diameter size in tenthsForeAftIce Contact 116th December,2013at 6:26:03,480.97.37.757011(open water)9(90-100)4(1.4-1.6 m)6(1.8-2.0 m)10(2000-5000 m)Ice Contact 222nd December,2013at 5:55:44,061.27.557.5590-4(60-70)6(80-90)0.6(1.0-1.2 m),2.1(1.2-1.4 m)1.5(1.4-1.6 m),0.8(1.6-1.8 m)4.3(2.0-2.5 m),0.7(2.5-3.0 m)5(20 m),3(20-100 m),2(100-500 m)*tenths means a percentage divided into ten parts.The mechanical properties of sample ice are shown in Tab.3,the sample ice was taken on 26thDecember when the vessel was waiting to be refueled.The thickness of sample ice was 1.5 m with0.5 m snow,and ice had a lot of porosity.Tab.3 Mechanical properties of 1.5 m ice during the voyageNumber of samplesMean valueStandard deviationMin.Max.Density/(kgm-3)Compressive strengthc/MPaHorizontal50.540.160.370.83813Vertical40.790.220.621.18839Flexural strengthf/kPa2149.921.9128.1171.7824The measured strain data on stern shoulder of both ice contacts are presented in Figs.3-4,respectively.(a)Shear strains from frame sensor(b)Normal strains from plate sensorNo.SS16 to No.SS23No.SS24 to SS29Fig.3 Strain data of Ice Contact 1第6期LIU Ying-hao et al:Determination of Ice Load on 909Ice Contact 1Ice Contact 1Shear strain(strain)Time(s)Time(s)Normal strain(strain)(a)Shear strains from frame sensor(b)Normal strains from plate sensorNo.SS16 to No.SS23No.SS24 to SS29Fig.4 Strain data of Ice Contact 22.2 Ice load determination on framesThe ice load acting on the stern frames was measured with V-shaped strain sensors installedon the frames.The shear strains were measured with strain sensors on the upper and lower part ofthe frame.The ice load is determined with the equation ofF=a (10)where F is the ice load vector,a is the influence coefficient matrix and is the vector of shearstrain difference between the two sensors on the same frame:=upper-lower(11)The matrix a is determined by the inverse influence coefficient matrix b,b=a-1=F-1(12)The element of the inverse coefficient matrix b is calculated bybij=j Fi(13)where j(j=1,n)is the shear strain difference measured on the frame j,the mounted frame number is n,andFiis the ice-induced load on the frame i.The elements of matrix b are established byapplying a calibration force F on each frame of stern shoulder,one frame at a time.Thereafter,thestrain difference could be identified each time.Then the influence coefficient matrix a could bedetermined with Eq.(12).For instance,a force of 1 kN is applied to Frame#39.5,and the response of each frame,j,could be identified with Eq.(11)when Frame#39.5 is loaded.Then each element of the inverse coefficient matrix b could be determined,and the matrix b related to the calibration force of 1 kN onFrame#39.5 is identified as follows:b11b12b13b14b21b22b23b24b31b32b33b34b41b42b43b440001=4140.54039.5(14)Hereafter,the influence coefficient matrix a could be determined by taking an inverse from thematrix b,and the actual ice load on frames of an ice-going vessel during a voyage could be identified with the full-scale measurement strain data according to Eqs.(10)-(11).910船舶力学第27卷第6期Ice Contact 2Ice Contact 2Time(s)Time(s)Shear strain(strain)Normal strain(strain)2.3 Ice load determination on a hull2.3.1 FE modelThe FE model used here has 250 432 nodes and 253 700 elements with frames from#36 to#44and was constructed with the 20 mm and 21 mm plates.The material of stern shoulder has the Poisson ratio=0.3,the Youngs moduli E=21011Pa and the shear moduli G=7.691010Pa.The model is presented from the inner and outer sides of the stern shoulder in Fig.5.The gray area in thecenter of the FE model indicates the area fitted with strain sensors,with extremely small meshing toobtain accurate results.Boundary conditions of the FE model are marked by blue triangles as shownin Fig.5.All the translational and rotational displacements are defined to be fixed at borders,andhave no impact on the results in the mounted area since the borders are set sufficiently far away.Toenhance the efficiency of the model analysis,some geometrical details of the vessel,for instance thelighting and opening holes of the frames are neglected to reduce the element number and shortenthe run time.(a)Inner side(b)Outer sideFig.5 Finite element model of stern shoulder2.3.2 Ice load discretizationUsually the classification regulations simplify the ice load area on a ship hull to be rectangular.However,in the reality,the load distribution is irregular,which has been found to be either line likeor having high pressure spots10-11.Thereby three discretizations are defined to depict the ice loaddistribution based on ice nature features,and the similar procedure has been utilized by Ikonen12.To enhance the precision of the inverse method,some hypotheses should be made as priori information13:(1)The ice load is presumed to be perpendicular to the structure since the shear forces causedby ice sheets sliding against the structure are ignored.(2)Assuming the positive direction of the ice load is into the structure,and the ice load is always nonnegative.(3)The contact area of ice load is presumed to be a strip of 0.8 m high,which is located at thecenter of the two platforms.Because during the voyage,the water line of the ship was observed to bemargi