Electronic Journal of Differential Equations, Vol. 2023 (2023), No. 15, pp. 1–24. ISSN: 1072-6691. URL: https://ejde.math.txstate.edu or https://ejde.math.unt.edu ESTIMATION OF PLATE PARAMETERS FROM VERTICAL DISPLACEMENT DATA USING A FAMILY OF PLATE MODELS LUTHER WHITE, TETYANA MALYSHEVA, LEIF KARLSTROM Abstract. We develop a method for estimation of parameters of an elastic plate resting on a Winkler-type elastic foundation solely from data on the vertical displacements of the plate. The method allows one to estimate com- ponents of the external body force density field, plate thickness, elastic foun- dation stiffness parameters, horizontal displacements of the plate, and stresses. The key idea of the method is that multiple plate models are used simulta- neously, namely the proposed reduced three-dimensional (R3D) plate model, the Mindlin plate model, and the thin plate model. The three plate models form a hierarchy of elastic plate models based on assumptions imposed on stresses, with the R3D plate model being the most generalized model and the thin plate model being the most constrained one. The hierarchical relationship among the plate models allows one to incorporate prior information into the estimation technique. The applicability of the proposed estimation method is illustrated by a numerical example. 1. Introduction We consider a three-dimensional elastic plate of non-uniform thickness resting on an elastic foundation and subjected to an external force. Our objective is to estimate the components of the external body force field, plate thickness, and the stiffness parameters of the foundation solely based on the measurements of the vertical displacement of the plate. The original motivation for our work arises from geophysics applications dealing with modeling and analysis of sill and laccolith deformation, where models of an elastic plate lying on an elastic foundation are used to capture the deflection of rock strata above a magma-filled intrusion in the subsurface [5, 12, 23]. In these applications, the goal is generally to infer the geometry of the intrusion, the force distribution generating deformation, and the material parameters of the surrounding rock, often solely from data on the earth surface. Magmatic intrusions exhibit a wide variety of shapes and depths, and often occur repeatedly over prolonged volcanic episodes. Plate models are an attractive framework to capture both isolated intrusions and sequences [24]. In fact, the problems of estimation of external forces and parameters for plate models have been of great practical interest in all fields of science and engineering where elastic plate models are employed. The identification of impact forces acting 2020 Mathematics Subject Classification. 35Q74, 74B05, 74K20, 35Q86, 86-10. Key words and phrases. Elastic plate model; linear elasticity; parameter estimation; force estimation; laccolith. ©2023. This work is licensed under a CC BY 4.0 license. Submitted February 28, 2022. Published February 17, 2023. 1 2 L. WHITE, T. MALYSHEVA, L. KARLSTROM EJDE-2023/15 on elastic plate structures - an important problem in impact engineering and struc- tural health monitoring - was studied in [11, 16, 20, 37, 38]. In [20], the iterative inversion method of deconvolution with Green’s functions of the medium was pre- sented to determine oblique force acting on a surface of an infinite plate. In [16], the impact load on a thick plate was determined inversely using theoretical Green’s function and simulated wave form under the assumption that the direction of the force is the only vertical. In [37], two methods - theoretical using Green’s functions and experimental - were developed to determine the impact-force history for thin plates based on the strain response. In [38] the transverse impact force history was determined for a rectangular Reissner-Mindlin plate based on the strain response using the eigenmode expansion method. In [11], the loading source of an infinite beam on an elastic foundation was inversely identified from given information of vertical deflection of the infinite beam using Tikhonov’s regularization. The work [11] was motivated by offshore hydrodynamic applications dealing with very large floating structures and ice plates in waves. Available literature on estimation of thickness of elastic plates is mostly limited to geoscience problems of estimation of the effective elastic thickness Te of Earth’s lithosphere modeled by thin elastic plates [3, 32]. Evaluation of Te is of great im- portance in geophysics as it has profound influence on the deformation and coupling of adjacent blocks within the theory of plate tectonics [3]. Common methods for es- timation of Te involve comparison of gravity anomalies and topography associated with vertical deflection of thin elastic plates [32]. To describe the plate-base interaction, we utilize a Winkler-type foundation model represented by a spring layer attached to the bottom of the plate and to the top of the rigid base. The Winkler foundation models are widely used in differ- ent fields of engineering to analyze beams, plates, and slabs resting on a soil medium [1, 28, 39]. The Winkler elastic foundations are also commonly employed in geo- physics in modeling of laccolith and sill formations to describe weak sedimentary layers along which laccolith or sill is emplaced [5, 12]. The behavior of the Winkler foundation is characterized by the spring constant called the subgrade modulus or the foundation stiffness parameter. Many researchers have made great efforts to evaluate the subgrade modulus [2, 5, 25, 31, 36, and the references therein]. The developed formulas are based on the elastic properties and geometry of the foundation. To our best knowledge, the present paper is the first one that attempts to esti- mate the external force field acting on an elastic plate, plate thickness, and elastic foundation stiffness parameters solely based on the measurements of the vertical displacement of the plate. The main difficulty here comes from the fact that a single plate model does not suffice to solve any of the given estimation problems. To overcome this obstacle, we propose the estimation method in which multiple plate models are used in conjunction. Specifically, we employ the family of three plate models resting on a Winkler-type elastic foundation: the thin plate model, the Mindlin plate model, and the proposed reduced three dimensional (R3D) plate model. The classical theory of elastic thin plates (see, for example, [13, 15]) assumes that comparatively small forces on plate surface are needed to bend it and there- fore, transverse shear and normal stresses are negligible. The Mindlin plate theory [13, 21], also known as the first-order shear deformation theory, takes into account EJDE-2023/15 ESTIMATION OF PLATE PARAMETERS 3 transverse shear effects, but retains zero transverse normal stress assumption and requires a shear correction factor. Although the Mindlin plate model provides suf- ficiently accurate results for moderately thick plates, it is not convenient to use due to the difficulty of determining an accurate shear correction factor [33]. The proposed R3D plate model does not impose zero stress assumptions existing in the thin and Mindlin plate models and does not involve a shear correction factor. At the same time, the R3D plate model preserves displacement assumptions embedded into the Mindlin plate model, thereby maintaining the simplicity of the first-order shear deformation theory. A distinguishing feature of the plate models under con- sideration is that, instead of stresses, the body force densities are used to capture external forces acting on the plate. It is assumed that the body force density field has a functional form compatible with that of the plate displacement field. The estimation method proposed in this paper involves the following steps. Ini- tially, the three plate models are used to establish relations among normal displace- ments and normal and tangential force densities. For comparison of the three plate models, given normal vertical displacement data, we first estimate normal force densities from each of the three plate models under the explicit constraining as- sumption that tangential forces are zero. Next, the three weighted plate models are used jointly to estimate both normal and tangential force densities, where forces are not constrained. It is assumed, however, that the actual tangential components of the body force are independent of the actual model being applied. Under a further assumption that the differences between predicted normal body force densities from the different models should be small, estimations of implied plate thickness and the foundation stiffness parameters are carried out. In addition, as a byproduct of our estimation technique, estimates of the stresses and horizontal displacements of the plate are also derived. Thus, the main contribution of this paper is the new and unique estimation method based on the simultaneous use of multiple elastic plate models, which allows one to estimate the components of an external force field acting on an elastic plate, plate thickness, elastic foundation stiffness parameters, as well as the stresses and horizontal displacements of the plate, solely from the measurements of the vertical displacement of the plate. This article is organized as follows. In Section 2, we present the distributed parameter models for an elastic plate under consideration. The models are for- mulated in a variational form based on the principle of minimum total potential energy. The variational formulation allows one to consider more general situations, for example, with discontinuities, and provides a foundation for finite element ap- proximations. In Section 3, the spatially discretized plate models are obtained by finite element approximations and represented by systems of algebraic equations. Ultimately, these are the models that are used in computations. In Section 4, we describe the techniques for estimation of model parameters from data on vertical displacement of the plate. In Section 5, we present a numerical example from the geoscience application to illustrate the applicability of the proposed method. 2. Plate model family In this section we provide variational formulations of three distributed parameter elastic plate models: the R3D plate, the Mindlin plate, and the thin plate, resting on the Winkler-type foundation. The variational formulations are the basis for finite 4 L. WHITE, T. MALYSHEVA, L. KARLSTROM EJDE-2023/15 element solutions and for our estimation of forces applied to an elastic medium from normal displacement data. The derivation of the R3D plate model is given in detail. Derivations of the Mindlin and thin plate models are well-known but are summarized to emphasize the differences. We begin by fixing the following notation to be used throughout this paper. In what follows, boldface letters denote vectors, both variables and functions. In par- ticular, i, j,k ∈ R3 are reserved for the basic unit vectors in Cartesian coordinates with the z-axis pointing upwards. Subscripts x, y, and z represent partial deriva- tives, and superscript T means transpose. Let Ωh = Ω × (−h2 , h 2 ) be a bounded open cylindrical domain in R3 representing a homogeneous in a vertical direction and isotropic elastic plate of variable thickness h. We assume that region Ω ⊂ R2 has a sufficiently smooth boundary ∂Ω and is large enough so that the plate un- dergoes negligible displacement on the boundary Γ = ∂Ω× (−h2 , h 2 ). We define the following spaces: H = L2(Ω), V0 = H1 0 (Ω), Hh = L2(Ωh,R3), Vh = H1(Ωh,R3) with standard norms, and Vh0 = { Φ ∈ Vh : Φ|Γ = 0 } with the norm ‖·‖Vh0 inherited from Vh. The symbol ‖·‖ stands for the Euclidean norm. Let the displacement at a point (x, y, z) ∈ Ωh be expressed as a vector-valued function u given by u(x, y, z) = u(x, y, z)i + v(x, y, z)j + w(x, y, z)k . (2.1) The displacement gradients are assumed to be small [9] so that higher-order powers of the displacements and their derivatives are neglected. Let ε = [εij ] 3 i,j=1 and τ = [τij ] 3 i,j=1 denote the strain and stress tensors, respectively. With the above as- sumptions, the strain-displacement relations for an isotropic material are expressed as ε = 1 2 (∇u +∇uT ) (2.2) and the stress-strain relations for an isotropic medium are given by τ = E 1 + ν [ ε+ ν 1− 2ν (trε)I ] , (2.3) where E is Young’s modulus, ν is Poisson’s ratio, trε is the trace of the tensor ε, and I is the identity tensor [6, 9]. We assume that E and ν may be spatially dependent. The elastic strain energy of the plate is defined to be VE(u) = 1 2 ∫ Ωh τ (u) : ε(u) dV . We assume that the plate is subjected to an external body force whose density is represented by F(x, y, z) = Fu(x, y, x)i + Fv(x, y, z)j + Fw(x, y, z)k . Then the work done on the plate by the above force is W(u) = ∫ Ωh F · u dV . (2.4) EJDE-2023/15 ESTIMATION OF PLATE PARAMETERS 5 A Winkler-type elastic foundation connecting the plate with a rigid foundation is characterized by positive functions κu(x, y), κv(x, y), and κw(x, y). These func- tions may be thought as spring coefficients indicating the stiffness of support from the elastic foundation. The potential energy functional associated with the elastic foundation is given by VF (u) = 1 2 ∫ Ω u(x, y,−h/2)TKFu(x, y,−h/2) dA , (2.5) where KF (x, y) = κu(x, y) 0 0 0 κv(x, y) 0 0 0 κw(x, y)  . (2.6) The total potential energy of the plate is then defined as Π(u) = VE(u) + VF (u)−W(u) . (2.7) To proceed further, we wish to express the total potential energy Π(u) in terms of functionals on Vh. The following assumption will be used throughout this article. Assumption 2.1. E, ν, κu, κv, κw,∈ L∞(Ω), and F ∈ Hh. Define the symmetric, continuous [27], bilinear form on Vh by αE(u,Φ) = ∫ Ωh τ (u) : ε(Φ) dV . Then the elastic strain energy can be expressed as VE(u) = 1 2 αE(u,u) . (2.8) Let γint 0 : Vh → H1/2(∂Ωh,R3) be a trace operator [27] defined by γint 0 (u) = u ∣∣ ∂Ωh , u ∈ Vh . Let ∂Ωb = Ω × {−h2 } denote a part of the boundary ∂Ωh corresponding to the bottom of the plate, and ∣∣ ∂Ωb be a restriction operator from H1/2(∂Ωh,R3) to H1/2(Ωb,R3). Then αF : Vh ×Vh → R given by αF (u,Φ) = ∫ Ω [( γint 0 (u) )∣∣ ∂Ωb ]T KF ( γint 0 (Φ) )∣∣ ∂Ωb dA (2.9) defines a symmetric, continuous, bilinear form on Vh. From (2.5) and (2.9), we have VF (u) = 1 2 αF (u,u) . (2.10) Finally, for every F ∈ Hh, we define a continuous linear functional f on Vh0 by the pairing Φ 7→ 〈f,Φ〉, where 〈f,Φ〉 = ∫ Ωh F ·Φ dV . (2.11) Applying (2.4), (2.8), (2.10), and (2.11) to (2.7) yields Π(u) = 1 2 αE(u,u) + 1 2 αF (u,u)− 〈f,u〉 . (2.12) Under assumption that the plate experiences negligible displacement on the boundary Γ, admissible displacements are u ∈ Vh0. According to the principle 6 L. WHITE, T. MALYSHEVA, L. KARLSTROM EJDE-2023/15 of minimum total potential energy, the displacement u that the region Ωh under- goes satisfies DΠ(u)Φ = 0, ∀ Φ ∈ Vh0 , (2.13) where DΠ(u)Φ = d dδ Π(u + δΦ) ∣∣∣ δ=0 is the first variation of Π at u. Applying (2.13) to (2.12), we arrive to the fol- lowing variational (weak) equation for an elastic plate resting on a Winkler-type foundation. αE(u,Φ) + αF (u,Φ) = 〈f,Φ〉, ∀ Φ ∈ Vh0 . (2.14) Recall Korn’s inequality [4, 13] that states that there exists a positive constant κ such that for any u ∈ Vh0 αE(u,u) ≥ κ‖u‖2Vh . (2.15) With (2.15) and αF (u,u) ≥ 0, the bilinear form (αE + αF ) on Vh is symmetric, continuous and Vh0-coercive. The Lax-Milgram theorem [30] applied to (2.14) immediately yields the well-posedness of the variational problem (2.14) associated with an elastic plate on a Winkler-type foundation. To obtain variational formulations of the R3D, Mindlin, and thin plate models, we specialize displacements (2.1) to various subspaces of displacement functions. Henceforth, for convenience, we set a = E (1 + ν)(1− 2ν) , â = E 1− ν2 , b = E 2(1 + ν) (2.16) and I1 = h, I2 = h2 4 , I3 = h3 12 . For the R3D plate model, we consider the components of the displacement vector u given by u(x, y, z) = zu1(x, y) v(x, y, z) = zv1(x, y) w(x, y, z) = w0(x, y) (2.17) with u1, v1, w0 ∈ V0. Here w0 is the displacement of the mid-surface of the plate in the z-direction, u1 and v1 are rotations of the normal to the mid-surface of the plate about the x-axis and the y-axis, respectively. Substituting (2.17) into (2.2) and (2.3) yields, respectively, the following strain-displacement and stress-displacement relations ε11 = zu1x, ε12 = z 2 (u1y + v1x), ε22 = zv1y, ε13 = 1 2 (u1 + w0x), ε33 = 0, ε23 = 1 2 (v1 + w0y), (2.18) and τ11 = za[(1− ν)u1x + νv1y], τ12 = zb(u1y + v1x), τ22 = za[νu1x + (1− ν)v1y], τ13 = b(u1 + w0x), τ33 = zνa[u1x + v1y], τ23 = b(v1 + w0y) . (2.19) EJDE-2023/15 ESTIMATION OF PLATE PARAMETERS 7 For compatibility with the assumed displacements, we consider the external force densities in the form Fu(x, y, z) = zfu(x, y) Fv(x, y, z) = zfv(x, y) Fw(x, y, z) = fw(x, y) (2.20) such that fu, fv, fw ∈ H. Here fu(x, y) and fv(x, y) are tangential force densities per unit depth. From (2.17) we observe that the displacement vector u is fully determined by the functions u1, v1, and w0. With this in mind, we proceed to derive the variational problem for the R3D plate model in terms of u1, v1, and w0. Applying (2.18)-(2.20) to (2.12) and integrating with respect to z, the total potential energy of the plate is expressed as Π(u1, v1, w0) = 1 2 ∫ Ω { aI3[(1− ν)u1x + νv1y]u1x + bI3(u1y + v1x)2 + bI1(u1 + w0x)2 + bI1(v1 + w0y)2 + aI3[νu1x + (1− ν)v1y]v1y + κuI2u 2 1 + κvI2v 2 1 + κwI1w 2 0 − 2[I3(fuu1 + fvv1) + I1fww0] } dA . Utilizing the principle of the minimum total potential energy (2.13) with Φ = zφ1(x, y)i + zψ1(x, y)j + ω0(x, y)k , (2.21) where φ1, ψ1, ω0 ∈ V0, and decoupling the result into individual variational equa- tions with respect to φ1, ψ1, and ω0, we arrive to the following variational formu- lation of the R3D plate model.∫ Ω { aI3[(1− ν)u1x + νv1y]φ1x + bI3(u1y + v1x)φ1y + bI1(u1 + w0x)φ1 + κuI2u1φ1 − I3fuφ1 } dA = 0 , (2.22) ∫ Ω { aI3[νu1x + (1− ν)v1y]ψ1y + bI3(u1y + v1x)ψ1x + bI1(v1 + w0y)ψ1 + κvI2v1ψ1 − I3fvψ1 } dA = 0 , (2.23) ∫ Ω { bI1(u1 + w0x)ω0x + bI1(v1 + w0y)ω0y + κwI1w0ω0 } dA− ∫ Σ I1fwω0 dA = 0 . (2.24) Next, we summarize the variational formulations of the Mindlin and thin plate models. For the Mindlin plate model, in addition to the displacement relations (2.17), it is assumed that in (2.3), τ33 = 0 (2.25) implying ε33 = −ν 1− ν (ε11 + ε22) . Moreover, a shear correction factor ks > 0 is introduced into the stress-strain relations (2.19) to account for the fact that shear strains are not constant over a 8 L. WHITE, T. MALYSHEVA, L. KARLSTROM EJDE-2023/15 cross-section of the plate [13]: τ13 = ksb(u1 + w0x), τ23 = ksb(v1 + w0y) . Thus, the total potential energy of the plate is expressed as Π(u1, v1, w0) = 1 2 ∫ Ω { âI3[u2 1x + 2νv1yu1x + v2 1y] + bI3(u1y + v1x)2 + ksbI1(u1 + w0x)2 + ksbI1(v1 + w0y)2 + κuI2u 2 1 + κvI2v 2 1 + κwI1w 2 0 − 2[I3(fuu1 + fvv1) + I1fww0] } dA . Applying the principle of the minimum total potential energy (2.13) with Φ in the form (2.21) and decoupling the result into individual variations give the variational formulation of the Mindlin plate model as follows.∫ Ω { âI3[u1x + νv1y]φ1x + bI3(u1y + v1x)φ1y + ksbI1(u1 + w0x)φ1 + κuI2u1φ1 − I3fuφ1 } dA = 0 ,∫ Ω { âI3[νu1x + v1y]ψ1y + bI3(u1y + v1x)ψ1x + ksbI1(v1 + w0y)ψ1 + κvI2v1ψ1 − I3fvψ1 } dA = 0 ,∫ Ω { ksbI1(u1 + w0x)ω0x + ksbI1(v1 + w0y)ω0y + κwI1w0ω0 − I1fwω0 } dA = 0 . For the thin plate model, it is further assumed that τ13 = τ23 = 0 . (2.26) This results in the conditions u(x, y, z) = −zw0x(x, y) v(x, y, z) = −zw0y(x, y) w(x, y, z) = w0(x, y) . The total potential energy of the plate is now Π(u1, v1, w0) = 1 2 ∫ Ω { âI3[(w0xx + w0yy)2 + 2(1− ν)(w2 0xy − w0xxw0yy)] + κuI3w 2 0x + κvI3w 2 0y + κwI1w 2 0 − 2[−I3(fuw0x + fvw0y) + I1fww0] } dA . Again, according to (2.13) applied with (2.21), the variational formulation of the thin plate model is given by∫ Ω { âI3(w0xx + w0yy)(ω0xx + ω0yy) + κuI3w0xω0x + κvI3w0yω0y + κwI1w0ω0 − [−I3(fuω0x + fvω0y) + I1fwω0] } dA = 0 , EJDE-2023/15 ESTIMATION OF PLATE PARAMETERS 9 where we have used the following result obtained from Green’s identities:∫ Ω { 2w0xyω0xy − w0xxω0yy − ω0xxw0yy } dA = 0 , for any ω0 ∈ H2 0 (Ω). 3. Spatial approximation In this section, spatially discretized plate models are determined from approxima- tions of the variational formulations for the R3D, Mindlin, and thin plate distributed parameter models. The cubic B-spline finite element method with enforcement of zero Dirichlet boundary conditions corresponding to the assumption that the plate undergoes negligible displacement on the boundary Γ is used. These systems are central in formulating estimation problems in which data consists only of vertical displacements at points within Ω. Let Sm(Ω) = {B1, . . . , Bm} (3.1) be a set of linearly independent functions in H2(Ω) from which we construct ap- proximations to the solutions of the model equations. Define the column m-vector- valued function B : Ω 7→ Rm by B(x, y) = [B1(x, y), . . . , Bm(x, y)]T , and express approximations of solutions u1, v1, and w0 and the force densities fu, fv, and fw as u1(x, y) = B(x, y)T ū1 v1(x, y) = B(x, y)T v̄1 w0(x, y) = B(x, y)T w̄0 (3.2) and fu(x, y) = B(x, y)T f̄u fv(x, y) = B(x, y)T f̄v fw(x, y) = B(x, y)T f̄w , (3.3) where ū1, v̄1, w̄0, f̄u, f̄v, and f̄w are m× 1 constant column vectors. Substituting (3.2) and (3.3) into the variational formulation (2.22)-(2.24) of the R3D plate model leads to{∫ Ω [ aI3(1− ν)BxB T x + bI3ByB T y + (bI1 + κuI2)BBT ] dA } ū1 + {∫ Ω [ aI3νBxB T y + bI3ByB T x ] dA } v̄1 + {∫ Ω bI1BBT x dA } w̄0 = {∫ Ω I3BBT dA } f̄u , (3.4) {∫ Ω [ aI3νByB T x + bI3BxB T y ] dA } ū1 + {∫ Ω [ aI3(1− ν)ByB T y + bI3BxB T x + (bI1 + κvI2)BBT ] dA } v̄1 + {∫ Ω bI1BBT y dA } w̄0 = {∫ Ω I3BBT dA } f̄v , (3.5) 10 L. WHITE, T. MALYSHEVA, L. KARLSTROM EJDE-2023/15{∫ Ω bI1BxB T dA } ū1 + {∫ Ω [ bI1ByB T ] dA } v̄1 + {∫ Ω [ bI1(BxB T x + ByB T y ) + κwI1BBT ] dA } w̄0 = {∫ Ω I1BBT dA } f̄w . (3.6) For the Mindlin plate model, we have similarly{∫ Ω [ âI3BxB T x + bI3ByB T y + (ksbI1 + κuI2)BBT ] dA } ū1 + {∫ Ω [ âI3νBxB T y + bI3ByB T x ] dA } v̄1 + {∫ Ω ksbI1BBT x dA } w̄0 = {∫ Ω I3BBT dA } f̄u , (3.7) {∫ Ω [ âI3νByB T x + bI3BxB T y ] dA } ū1 + {∫ Ω [ âI3ByB T y + bI3BxB T x + (ksbI1 + κvI2)BBT ] dA } v̄1 + {∫ Ω ksbI1BBT y dA } w̄0 = {∫ Ω I3BBT dA } f̄v , (3.8) {∫ Ω ksbI1BxB T dA } ū1 + {∫ Ω ksbI1ByB T dA } v̄1 + {∫ Ω ksbI1(BxB T x + ByB T y ) + κwI1BBT dA } w̄0 = {∫ Ω I1BBT dA } .f̄w . (3.9) Finally, for the thin plate model, we obtain{∫ Ω [ âI3(Bxx + Byy)(Bxx + Byy)T + I3(κuBxB T x + κvByB T y ) + κwI1BBT ] dA } w̄0 = {∫ Ω −I3BxB T dA } f̄u + {∫ Ω −I3ByB T dA } f̄v + {∫ Ω I1BBT dA } f̄w . (3.10) To allow for the spatial variability of the coefficients and plate thickness, let {Ωk}Nk=1 be a partition of Ω, Ω = ⋃ k Ωk, and define the functions Ξk(x, y) = { 1, when (x, y) ∈ Ωk, 0, when (x, y) /∈ Ωk . We express the coefficients (2.6) and (2.16) and plate thickness h as linear com- binations of Ξk of the form θ(x, y) = N∑ k=1 θkΞk(x, y) EJDE-2023/15 ESTIMATION OF PLATE PARAMETERS 11 where θ represents a, â, b, κu, κv, κw, or h, whichever is appropriate. Note that, for example, h(x, y)a(x, y) = N∑ k=1 hkakΞk(x, y) . It is convenient to define the following matrices. G (k) 0 = ∫ Ω ΞkBBT dA with G0 = N∑ k=1 G (k) 0 and G00(θ) = N∑ k=1 θkG (k) 0 , and G (k) x0 = ∫ Ω ΞkBxB T dA, G (k) y0 = ∫ Ω ΞkByB T dA G(k) xx = ∫ Ω ΞkBxB T x dA, G(k) yy = ∫ Ω ΞkByB T y dA G(k) xy = ∫ Ω ΞkBxB T y dA G (k) L = ∫ Ω Ξk(Bxx + Byy)(BT xx + BT yy) dA with Gx0(θ) = N∑ k=1 θkG (k) x0 , Gy0(θ) = N∑ k=1 θkG (k) y0 Gxx(θ) = N∑ k=1 θkG (k) xx , Gyy(θ) = N∑ k=1 θkG (k) yy Gxy(θ) = N∑ k=1 θkG (k) xy , GL(θ) = N∑ k=1 θkG (k) L , where we express dependence on the coefficients when necessary. Finally, with reference to (3.4)-(3.6), (3.7)-(3.9), and (3.10), we define the following matrices for the R3D plate, Mindlin plate, and the thin plate models, respectively. For ease of exposition, subscripts R, M , and T will refer to the respective plate models. For the R3D plate model, we introduce the matrices GR11 = Gxx((1− ν)I3a) +Gyy(I3b) +G00(I1b) +G00(I2κu) GR12 = Gxy(νI3a) +Gyx(I3b) GR13 = G0x(I1b) GR22 = Gyy((1− ν)I3a) +Gxx(I3b) +G00(I1b) +G00(I2κv) GR23 = G0y(I1b) GR33 = Gxx(I1b) +Gyy(I1b) . (3.11) 12 L. WHITE, T. MALYSHEVA, L. KARLSTROM EJDE-2023/15 For the Mindlin plate model, we define the matrices GM11 = Gxx(I3â) +Gyy(I3b) + ksG00(I1b) +G00(I2κu) GM12 = Gxy(νI3â) +Gyx(I3b) GM13 = ksG0x(I1b) GM22 = Gyy(I3â) +Gxx(I3b) + ksG00(I1b) +G00(I2κv) GM23 = ksG0y(I1b) GM33 = ksGxx(I1b) + ksGyy(I1b) . (3.12) For the thin plate, we assign the matrix HT = GL(I3â) +Gxx(I3κu) +Gyy(I3κv) +G00(I1κw) . (3.13) Of course, terms may be written as multipliers of the matrices in the case of constant thickness, Young’s modulus, and Poisson’s ratio. Applying (3.11) to (3.4)-(3.6) renders the following system for the R3D plate model. GR11ūR1 +GR12v̄R1 +GR13w̄R0 = G0(I3)f̄Ru GTR12ūR1 +GR22v̄R1 +GR23w̄R0 = G0(I3)f̄Rv GTR13ūR1 +GTR23v̄R1 +GR33w̄R0 = G0(I1)f̄Rw . Similarly, applying (3.12) to (3.7)-(3.9), for the Mindlin plate model we obtain the system GM11ūM1 +GM12v̄M1 +GM13w̄M0 = G0(I3)f̄Mu , GTM12ūM1 +GM22v̄M1 +GM23w̄M0 = G0(I3)f̄Mv , GTM13ūM1 +GTM23v̄M1 +GM33w̄M0 = G0(I1)f̄Mw . Observe that the R3D and Mindlin systems have the form G∗11ū∗1 +G∗12v̄∗1 +G∗13w̄∗0 = G0(I3)f̄∗u , GT∗12ū∗1 +G∗22v̄∗1 +G∗23w̄∗0 = G0(I3)f̄∗v , GT∗13ū∗1 +GT∗23v̄∗1 +G∗33w̄∗0 = G0(I1)f̄∗w , (3.14) where ∗ = R or M , whichever is appropriate. Next, we wish to derive the expressions for w̄∗0 in terms of f̄∗u, f̄∗v, and f̄∗w. To this end, we introduce the matrices H∗11 = G∗22 −GT∗12G −1 ∗11G∗12 H∗12 = G∗23 −GT∗12G −1 ∗11G∗13 H∗22 = G∗33 −GT∗13G −1 ∗11G∗13 H∗ = H∗22 −HT ∗12H −1 ∗11H∗12 and vectors f̃∗v = G0(I3)f̄∗v −GT∗12G −1 ∗11G0(I3)f̄∗u , (3.15) f̃∗w = G0(I1)f̄∗w −GT∗13G −1 ∗11G0(I3)f̄∗u . (3.16) Then the solution of (3.14) may then be expressed as H∗w̄∗0 = f̃∗w −HT ∗12H −1 ∗11f̃∗v , (3.17) H∗11v̄∗1 = f̃∗v −H∗12w̄∗0 , (3.18) EJDE-2023/15 ESTIMATION OF PLATE PARAMETERS 13 G∗11ū∗1 = G0(I3)f̄∗u −G∗12v̄∗1 −G∗13w̄∗0 . (3.19) Applying (3.15) and (3.16) to (3.17), we obtain the following expression for w̄∗0 for the R3D and the Mindlin plate models, H∗w̄∗0 = ( HT ∗12H −1 ∗11G T ∗12 −GT∗13 ) G−1 ∗11G0(I3)f̄∗u −HT ∗12H −1 ∗11G0(I3)f̄∗v +G0(I1)f̄∗w . (3.20) From (3.10) and (3.13), for the thin plate model we have HT w̄T0 = Gx0(I3)f̄Tu +Gy0(I3)f̄Tv +G0(I1)f̄Tw . (3.21) 4. Estimation of parameters In this section, we consider the estimation of body force densities, plate thickness, and elastic foundation stiffness parameters based on vertical displacement data. Specifically, we assume that data of the vertical displacements zi at points (xi, yi) ∈ Ω for i = 1, . . . , No are given. The first problem is to estimate the force density components f̄u, f̄v, and f̄w under the assumption that the values of plate thickness h and elastic foundation stiffness parameters κu, κv, and κw are specified. A related problem is the interpolation problem [26, 35] that seeks a function z(x, y) defined for (x, y) ∈ Ω such that z(xi, yi) = zi. Towards this end, we express the function z as z(x, y) = B(x, y)T z̄ , where the components of B(x, y) are a set of linearly independent functions in Sm (3.1). We seek to determine the m-vector z̄ to minimize the error functional EI(z̄) = No∑ i=1 (B(xi, yi) T z̄− zi)2 . Define the interpolation matrix J = No∑ i=1 B(xi, yi)B(xi, yi) T and the vector ζ̄ = No∑ i=1 ziB(xi, yi) . Under the assumption that the data is sufficient that J is invertible, the interpo- lating function obtained for the data (xi, yi, zi) is given by z(x, y) = B(x, y)TJ−1ζ̄ (4.1) The objective is to use the interpolating function (4.1) in conjunction with the plate models to obtain information on the body force densities. The underlying as- sumption is that the vertical displacement and the forces densities are independent of the model. Accordingly, we set w0(x, y) = z(x, y) Hence, with w0(x, y) = B(x, y)T w̄0 it follows from linear independence of the components of B(x, y) that w̄0 = J−1ζ̄ . 14 L. WHITE, T. MALYSHEVA, L. KARLSTROM EJDE-2023/15 With reference to (3.20) and (3.21), we define the matrices N∗u = ( HT ∗12H −1 ∗11G T ∗12 −GT∗13 ) G−1 ∗11G0(I3) N∗v = −HT ∗12H −1 ∗11G0(I3) NTu = Gx0(I3) NTv = Gy0(I3) . Then the relations (3.20) and (3.21) yield the following set of constraining equations among the quantities w̄0, f̄u, f̄v, and f̄w. NTuf̄u +NTv f̄v +G0(I1)f̄w = HT w̄0 , (4.2) NMuf̄u +NMv f̄v +G0(I1)f̄w = HM w̄0 , (4.3) NRuf̄u +NRv f̄v +G0(I1)f̄w = HRw̄0 . (4.4) Remark 4.1. Under the assumption that the tangential forces are relatively small, setting f̄u = f̄v = 0 (4.5) from (4.2)-(4.4) for the thin plate, the Mindlin plate, and the R3D plate models, respectively, we have f̄∗w = [G0(I1)]−1H∗w̄0 , (4.6) where ∗ stands for T , M , and R, respectively. Equation (4.6) will be used in Section 5 to compare numerical estimates of normal force densities produced by each of the three plate models under assumption that the tangential external body force is zero. Continuing on subtracting (4.2) from (4.3) and (4.4) and defining the matrices LMu = NMu −NTu, LMv = NMv −NTv LRu = NRu −NTu, LRv = NRv −NTv , we have LMuf̄u + LMv f̄v = (HM −HT )w̄0 (4.7) LRuf̄u + LRv f̄v = (HR −HT )w̄0 (4.8) as conditions from the models to constrain the in-plane body force densities f̄u and f̄v. In practice, system (4.7) and (4.8) often involves ill-conditioned matrices. To overcome this obstacle, we seek a solution (f̄u, f̄v) of (4.7) and (4.8) by minimizing the quadratic error functional E(f̄u, f̄v) = σM‖LMuf̄u + LMv f̄v − (HM −HT )w̄0‖2 + σR‖LRuf̄u + LRv f̄v − (HR −HT )w̄0‖2 + σu‖f̄u‖2 + σv‖f̄v‖2 , (4.9) where σM , σR, σu, and σv are positive weights to be chosen to reflect estimates of the precision of the corresponding term. For example, in computations in Section 5, we weight the variances for the R3D and the Mindlin the same by choosing σR and σM both equal 1 while σu and σv both equal to 10−16 to reflect the previous estimates when f̄u and f̄v were assumed to be zero. EJDE-2023/15 ESTIMATION OF PLATE PARAMETERS 15 Remark 4.2. The R3D plate, Mindlin plate, and thin plate models form a hier- archy of plate models based on assumptions imposed on stresses. The R3D plate model represents the most generalized (top) plate model with the only assump- tion on stresses given by the stress-strain relation (2.3). The R3D plate model is followed by the Mindlin plate model that carries zero transverse normal stress assumption (2.25). The Mindlin plate model is in turn followed by the thin plate model that imposes additional assumption that transverse shear stresses are negli- gible (2.26). With this in mind, the estimates (4.6) obtained under constraint (4.5) can be considered as prior information from each individual plate model. Then the minimization of E(f̄u, f̄v) corresponds to finding the maximum likelihood estima- tor of a probability density function that includes priors from the individual plate models. Next, we introduce the matrices Luu = σML T MuLMu + σRL T RuLRu + σuI Lvv = σML T MvLMv + σRL T RvLRv + σvI Luv = σML T MuLMv + σRL T RuLRv , where I is the identity matrix, and the vectors WT u = σM [ (HM −HT )w̄0 ]T LMu + σR [ (HR −HT )w̄0 ]T LRu WT v = σM [ (HM −HT )w̄0 ]T LMv + σR [ (HR −HT )w̄0 ]T LRv With the above notation, the error functional (4.9) takes the form E(f̄u, f̄v) = f̄Tu Luuf̄u + 2f̄Tu Luv f̄v + f̄Tv Lvv f̄v − 2 [ WT u f̄u + WT v f̄v ] + E(0, 0) . The minimizer (f̄u, f̄v) of E satisfies the normal equations Luuf̄u + Luv f̄v = Wu LTuv f̄u + Lvv f̄v = Wv which yield f̄v = [ Lvv − LTuvL−1 uuLuv ]−1( Wv − LTuvL−1 uuWu ) f̄u = L−1 uu ( Wu − Luv f̄v ) . Equations (4.2)-(4.4) now render f̄∗w = [G0(I1)]−1 ( H∗w̄0 −N∗uf̄u −N∗v f̄v ) , (4.10) where ∗ corresponds to T , M , and R, respectively. Estimates for ū∗1 and v̄∗1 are obtained by means of equations (3.15)-(3.19). Using (3.2), we obtain the tangential components of displacement, and from the stress-displacement relations (2.19) of the R3D plate model we obtain strain estimates. Finally, we make the following observation. Remark 4.3. The continuous dependence of force densities, displacements, and stresses on parameters h, κu, κv, and κw follows from the algebraic expressions satisfied by the matrices and vectors in the above. Remark 4.4. Expressions for estimates of of body force densities, displacements, and stresses are obtained from normal displacement data given plate thickness h and foundation stiffness parameters κu, κv, and κw under the assumption that 16 L. WHITE, T. MALYSHEVA, L. KARLSTROM EJDE-2023/15 the body force density components are independent of the model. The three model equations are used in essence to solve simultaneously for f̄u and f̄v. Different models then yield different estimates for the normal body force density component f̄w, even though the estimates may be quite close for certain values of h, κu, κv, κw (see, for instance, the numerical example in Section 5). Next, we consider the estimation of the plate thickness h and foundation stiffness parameters κu, κv, and κw under the assumption that the estimations of f̄w from different plate models should be close. To that end, we set q = (h, κu, κv, κw) and proceed, for simplicity, as though they are constants. The case in which the parameters are variable is similar. Let Q ⊂ R4 be a set of admissible parameters q. We consider the cost functional measuring the total difference between the estimates of f̄w from pairs of different plate models in the form Jc(q) = ξRT ‖f̄Rw(q)− f̄Tw(q)‖+ ξMT ‖f̄Mw(q)− f̄Tw(q)‖ + ξRM‖f̄Rw(q)− f̄Mw(q)‖ (4.11) and seek the value q∗ ∈ Q minimizing the above functional. The estimates f̄∗w(q), with ∗ = R, T , and M , are obtained from (4.10), and the positive weights ξRT , ξMT , and ξRM are chosen to emphasize a particular model. Typically, we choose ξRT = ξMT = ξRM . The physical meaning of parameters q suggests that Q is compact in R4. There- fore, from continuity of the mapping given in terms of the functional (4.11), there exists the minimizer of Jc(q). In the next section we present the results for this minimization. 5. Numerical example We present a numerical example motivated by the geoscience application of de- formation associated with shallow, sill-like magma intrusions (laccolith formation). A classic and well-studied example of laccolith formation is the Henry Mountains, UT, the last-named mountain range in the continental USA. The Henry Mountains were formed by a complex of shallow intrusions 28-25 million years ago [22] at 1.5- 2 km depths. Deformation of overlying sandstone layers by the intrusions is well documented due to erosion into the tops of the intrusions (see, for example [10]). The vertical displacement data are obtained from the structure contour map of Hunt et al. [8] for Mt. Holmes and Mt. Ellsworth in Henry Mountains, noting the top-most ∼100 m is interpolated. Mt. Holmes and Mt. Ellsworth topographic map and Henry Mountains structure contour map with interpolated top are presented in Figure 1 (A) and (B), respectively. The plate thickness h, size of the domain Ω, and elastic parameters E and ν are given as geophysically reasonable numbers. The foundation stiffness parameters κu, κv, and κw and the shear correction factor ks are specified through numerical experience. For the sake of simplicity we assume κu = κv = κw and suppress sub- scripts u, v, and w. The values of parameters used in the example are summarized in Table 1. To develop a spatial approximation, we consider Ω = (0, Lx) × (0, Ly). For the approximation basis functions we use products of cubic B-splines defined on uniform partition on (0, Lx) and (0, Ly), with nx and ny subintervals, respectively EJDE-2023/15 ESTIMATION OF PLATE PARAMETERS 17 (A) (B) Figure 1. (A) Topographic map of Mt. Holmes (top) and Mt. Ellsworth (bottom), Henry Mountains, UT. (B) Structure contour map with interpolated top-most ∼100 m (B). Table 1. Values of parameters used in the example Plate horizontal dimensions Lx = 14, 600 m, Ly = 15, 000 m Number of subintervals of partitions nx = ny = 12 Plate thickness h = 1400 m Young’s modulus E = 1.0× 109 Pa Poisson’s ratio ν = 0.25 Foundation stiffness parameter κ = 105 N/m3 Shear correction factor ks = 1 Weights of the models σM = σR = 1, σu = σv = 10−16 ξRT = ξMT = ξRT = 1 [26]. Accordingly, let b (x) i (x), for i = 1, . . . ,mx = nx + 3 and b (y) i (y), for i = 1, . . . ,my = ny + 3 denote cubic B-splines defined with respect to the meshes. In addition, define the mx- and my-column-vector-valued functions by b(x)(x) = [b (x) 1 (x), . . . , b(x) mx (x)]T b(y)(y) = [b (y) 1 (y), . . . , b(y) my (y)]T Finally, define the m = mxmy-column vector as the tensor product of b(x) and b(y) B(x, y) = b(x)(x)⊗ b(y)(y) 18 L. WHITE, T. MALYSHEVA, L. KARLSTROM EJDE-2023/15 Results in the graphs are presented on the domain (0, 104)× (0, 1.5× 104) m2 to account for edge affects. The estimates of the scalar components of displacements and force densities are obtained from (3.2) and (3.3). As before, the subscripts R, M , and T stand for the R3D, Mindlin, and thin plate model, respectively. Uplift data are given at (x, y) locations over the domain, and the Henry Mountains surface data are shown in Figure 2(A). (A) (B) Figure 2. (A) Henry Mountains surface data. (B) Force density fRw estimated from the R3D plate model with fu = fv = 0. The graph of the normal body force density fRw using the R3D plate model with fu = fv = 0 is portrayed in Figure 2(B). The estimates of f̄Rw are calculated from (4.10). For comparison with the thin plate and Mindlin plate models, relative differences in estimates of fw from different models are given below in which the absolute difference is normalized by the H-norm of fRw. ‖fRw − fTw‖H ‖fRw‖H = 1.44×10−4 ‖fMw − fTw‖H ‖fRw‖H = 1.43×10−4 ‖fRw − fMw‖H ‖fRw‖H = 7.88×10−6 . The estimated tangential body force densities per unit depth, fu and fv, obtained from the family of the three plate models are presented in Figures 3 (A) and (B), respectively. The corresponding values of f̄u and f̄v are minimizers of the error functional (4.9). One can observe that both fu and fv are relatively small compared to the result for fRw as to be expected. Given the estimated fu and fv, the normal force density fw may be estimated from the equations (4.10) for each plate model. Figure 3(C) represents the normal force density fRw obtained from (4.10) using the R3D plate model. As one can observe from the graphs in Figures 2(B) and 3(C), the estimates of fRw with and without assumption that fu = fv = 0 are quite close, since fu and fv are relatively small for the values of model parameters used in this example. The graph of the relative difference between the fRw estimated from the family of three plate models without assumption fu = fv = 0 and fRw obtained from the R3D plate model under the assumption fu = fv = 0 is given in Figure 3(D). EJDE-2023/15 ESTIMATION OF PLATE PARAMETERS 19 (A) (B) (C) (D) Figure 3. Force densities estimated from the family of plate mod- els: (A) Force density per unit depth, fu. (B) Force density per unit depth, fv. (C) Force density fRw. (D) Relative difference between fRw in Figures 2 (B) and 3 (C). For comparison with the thin and Mindlin plate models, below we present relative differences in estimates of fw calculated from (4.10) for different models without assumption that fu = fv = 0. ‖fRw − fTw‖H ‖fRw‖H = 2.7×10−3 ‖fMw − fTw‖H ‖fRw‖H = 3.0×10−3 ‖fRw − fMw‖H ‖fRw‖H = 3.53×10−4 . Figures 4 (A) and (B) show the estimated tangential displacement components u and v obtained from (3.15)-(3.19) for the R3D plate model. Here the tangential force densities per unit depth, fu and fv, are estimated from the family of three plate models and fw is calculated from (4.10) for the R3D plate model. The values of u and v estimated from the R3D plate model are used to calculate the transverse shear and normal stresses τ13, τ23, and τ33 from the strain-displacement and stress- strain relations (2.2) and (2.3), respectively. The calculation results are pictured in Figures 4 (C)–(E). 20 L. WHITE, T. MALYSHEVA, L. KARLSTROM EJDE-2023/15 (A) (B) (C) (D) (E) Figure 4. Displacements and stresses estimated from the R3D plate model with fu and fv estimated from the family of plate models and subsequent estimation of fw from the R3D plate model: (A) Displacement u. (B) Displacement v. (C) Stress τ13. (D) Stress τ23. (E) Stress τ33. Figure 5(A) represents the level curves of the cost functional Jc (4.11) measuring total difference between the estimates of the normal force density fw from the pairs of different plate models as a function of parameters q = (h, κ). The direct evalua- tion of Jc gives the unique global minimum, with respect to the chosen evaluation mesh, at h = 2000 m and κ = 3 × 106 N/m3. Figure 5(B) shows the absolute EJDE-2023/15 ESTIMATION OF PLATE PARAMETERS 21 (A) (B) (C) Figure 5. Optimal values of the plate thickness h and the foun- dation stiffness parameter κ: (A) The level curves of the cost func- tional Jc(h, κ). (B) Cost functional Jc as a function of h with optimal κ = 3 × 106 N/m3. (C) Cost functional Jc as a function of κ with optimal h = 2000 m. value of the cost functional Jc as a function of h, while the foundation stiffness parameter κ is fixed at the optimal value 3×106 N/m3. Figure 5(C) represents the absolute value of the cost functional Jc as a function of κ, with the optimal plate thickness h = 2000 m fixed. For easy representation, in Figures 5 (B) and (C), the value of the cost functional Jc is rescaled by dividing its absolute value by Young’s modulus E. According to Jackson and Pollard [10], the intrusions exposed in the Henry Mountains were emplaced at ∼2-4 km depth. In [22], the estimated depth of the intrusion is ∼1.5-2 km. Work [5] that employs a Winkler elastic foundation to represent a weak sedimentary layer, such as clay, tuff, or unconsolidated sediments along which laccolith and sill intrusions take place, provides a general geological range for the foundation stiffness parameter κ = 104 − 109 N/m3. Thus, the ob- tained estimates of the plate thickness h = 2000 m and the foundation stiffness parameter κ = 3 × 106 excellently fit their geophysically reasonable ranges which demonstrates the applicability of the proposed estimation method. 22 L. WHITE, T. MALYSHEVA, L. KARLSTROM EJDE-2023/15 6. Conclusions In this paper we presented a novel and unique method for estimation of pa- rameters of an elastic plate resting on a Winkler-type foundation solely from the measurements of the vertical displacement of the plate. The method is based on the multiple plate models used in conjunction and allows one to estimate components of the external body force density field, plate thickness, elastic foundation stiffness parameters, as well as horizontal displacements of the plate and stresses. Since neither a single plate model nor a pair of plate models suffices to estimate any of the above mentioned parameters, we employed a family of three plate models - the proposed R3D plate model, the Mindlin plate model, and the thin plate model. The R3D plate model follows the displacement assumptions embedded into the Mindlin plate model, but does not impose zero stress assumptions existing in the thin and Mindlin plate models and does not involve a shear correction factor from the Mindlin plate model. Thus, the three plate models under consideration form a hierarchy of elastic plate models based on assumptions imposed on stresses, with the R3D plate model being the most generalized model and the thin plate model being the most constrained one. This hierarchical relationship among the plate models serves as a source of prior information in the estimation method. The main advantage and important practical benefit of the proposed method is that only data on vertical displacement are needed to estimate the external body force density field acting on an elastic plate, plate thickness, and foundation stiffness parameters. In addition, as a byproduct of this estimation technique, we also obtain estimates of the stresses and horizontal displacements of the plates. The method has the wide range of applications since estimation of external forces and parameters of plate models are of great practical interest in all fields of science and engineering where elastic plates are employed. The applicability of the proposed estimation method is illustrated by a numerical example motivated by geoscience applications dealing with modeling and analysis of laccolith formation. Acknowledgments. The authors are grateful to Dr. Daniel O’Hara for his assis- tance with the Henry Mountains data. L. 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White Oregon Applied Mathematics Institute, 2130 Ridgeway Dr., Eugene, OR 97401, USA Email address: oregami.lwhite@gmail.com Tetyana Malysheva Department of Mathematics & Statistics, Resch School of Engineering, University of Wisconsin-Green Bay, 2420 Nicolet Drive, Green Bay, WI 54311, USA Email address: malyshet@uwgb.edu Leif Karlstrom Department of Earth Sciences, 1272 University of Oregon, Eugene, OR 97403, USA Email address: leif@uoregon.edu 1. Introduction 2. Plate model family 3. Spatial approximation 4. Estimation of parameters 5. Numerical example 6. Conclusions Acknowledgments References