A shell is a three-dimensional solid whose thickness in one dimension is considerably small relative to the other two dimensions. The mathematical model of a shell can be reduced to a two-dimensional one by considering only the mechanics on some reference surface (see Fig.  for an illustration).

One typically distinguishes between thick- and thin-shell models. Thick-shell models capture transverse shear strains in addition to membrane and bending strains as opposed to thin-shell models, where the shell thickness is assumed to be small enough so that the effects of transverse shear deformations can be neglected. The configuration of a thin shell is fully determined by the position of its reference surface in physical space, whereas the configuration of a thick shell is supplemented by a deformable vector field on the reference surface called a director field.

To showcase the capabilities of isogeometric analysis in solving higher-order problems numerically, we focus on the displacement formulation of the Koiter model for thin shells and the Kirchhoff model for thin plates, which require C¹ finite elements for a conforming discretization. Depending on the ratio of the shell thickness to the scale of the simulation, a thick-shell model might be more adequate for capturing the correct behavior of the lithosphere.

We model the lithosphere as a thin elastic plate of effective elastic thickness t floating on the asthenosphere and subject to gravitational forces (see Fig. 4a for an illustration). The initial depth of the mid-surface in the undeformed configuration corresponds to the theoretical depth relative to the mean sea level when there is no overlying mass. The actual mid-surface of the lithosphere does not have to coincide with the Mohorovičić surface, which is the boundary between the crust and the upper mantle of the Earth. However, we assume that they are close to each other and differ only by a constant vertical displacement, which is a simplification of the model that disregards subsurface variations, usually obtained via inversion of gravity anomalies.

Starting from the equilibrium equations for a Kirchhoff plate in Sect.  with the external load fext, we split up the contributions from gravity and buoyancy: fext=fgrav+fbuoy.

Gravitational load is obtained by integrating all of the weight above the mid-surface. The density of overlying air is considered to be negligible so that the weight of topography ranges from Earth's surface down to the mid-surface. It is given by 17fgrav=-∫dhϱgdz, where d is the depth of the mid-surface relative to the mean sea level; h is the topographic elevation; ϱ is the density of overlying mass; and g is the gravitational acceleration, which is assumed to be constant for the sake of simplicity.

The vertical displacement of the mid-surface from the initial depth is given by w=d-d0 (see Fig. 4b). The buoyant force is equal to the weight of the displaced asthenosphere; thus 18fbuoy=∫dd0ϱmgdz=-ϱmgw, assuming a constant upper-mantle density ϱm.

Instead of working with the actual topographic elevation, we use a mass representation r obtained by taking the mass above the mid-surface of the lithosphere and normalizing it by some depth-independent reference density ϱr. We choose the reference density as the mean rock density from the current depth of the mid-surface to its initial depth and assume that it is homogeneous in space so that 19ϱr=1|w|∫dd0ϱdz,r=1ϱr∫d0hϱdz.

Using the mass representation, which corresponds to rock-equivalent topography, we can write the external load as 20fext=-(ϱm-ϱr)gw-ϱrgr. Plugging this into the weak formulation for the plate model without external boundary forces yields a(w,v)+b(w,v)=c(v) with the bilinear form in Eq. () and 21b(w,v)=∫A(ϱm-ϱr)gwvdA,c(v)=-∫AϱrgrvdA. This is the Vening-Meinesz model of flexural isostasy used to explain regional compensation (, Chap. 5).

In the case where there is no flexural rigidity, the Vening-Meinesz model is reduced to the Airy–Heiskanen model of local isostasy , for which the well-known relation 22ϱm-ϱrϱr(d0-d)=r holds. It states that the lithospheric depression relative to the initial depth is proportional to the mass representation of the topography with a scaling factor of (ϱm-ϱr)/ϱr. Using the above relation, we can determine the initial depth from some standard crustal thickness t0 corresponding to a lithospheric plate in local isostasy when the topographic elevation is zero.

If we consider only a portion of Earth's lithosphere for the simulation, conditions on the boundary of the domain have to be prescribed to compensate for the missing information outside of it. A natural choice is given by the full Neumann boundary condition, which corresponds to setting gext=0 on the whole boundary. The resulting isostatic boundary value problem for the plate model then reads as follows: find an admissible deflection w such that a(w,v)+b(w,v)=c(v) for all admissible variations v=δw.

The Sobolev space H2(A) is chosen as the space of admissible deflections for the boundary value problem. It consists of square-integrable functions on the reference surface with square-integrable weak derivatives up to the second order. For the full Neumann problem, the variational problem is well-posed by the Lax–Milgram theorem (, Chap. II), provided that A⊂R2 is a bounded Lipschitz domain and the coefficients D(1-ν) and (ϱm-ϱr)g are bounded from below by a positive number. The H2 coercivity of the bilinear form follows from the fact that the H2 norm is equivalent to a similar one without the terms containing first-order derivatives (, Theorem 1.8).

The above displacement formulation requires H2 regularity, which implies global C¹ continuity for the trial and test functions. The difficulty of C¹ finite elements can be circumvented by considering isogeometric shape functions.

Using the more general shell equations, it is possible to perform simulations of the lithosphere on the whole surface of the Earth. From a modeling point of view, the results may not reflect the physical reality since the Earth consists of different regimes and tectonic plates that interact with each other in a complex manner. Furthermore, due to the large scale of the simulation, the effects of flexural rigidity will not be visible. Nevertheless, we assume that the entire lithosphere can be modeled as a single spherical shell to showcase the capabilities of isogeometric analysis in numerical simulations on curved domains, especially a spherical domain. Note that it is also possible to model the surface of the Earth as an oblate spheroid or an irregular geoid instead of a sphere using isogeometric analysis. For the sake of simplicity, we restrict ourselves to the spherical model in this paper.

Some considerations in Sect.  for lithospheric plates in isostatic equilibrium have to be adapted to the shell model. The buoyant force in three dimensions reads as 23b(u,v)=∫A(ϱm-ϱr)g(n⋅u)(n⋅v)dA, where (n⋅u) and (n⋅v) are the radial parts of the trial and test functions, respectively, given by the orthogonal projection onto the unit normal n of the sphere. Similarly, the external load is given by a radial gravitational force 24c(v)=-∫Aϱrgr(n⋅v)dA. With the above adjustments, the isostatic problem for a Koiter shell then reads as follows: find an admissible displacement u such that a(u,v)+b(u,v)=c(v) holds for all admissible variations v=δu. We consider the vector-valued Sobolev space H2(A)3 for the displacements of the spherical shell.

Let ϑ0≤⋯≤ϑm be a finite sequence of non-decreasing real numbers. A spline of degree p is a piecewise polynomial function f:[ϑ0,ϑm]→R, with the property that the restriction to each subinterval [ϑi-1,ϑi) for i=1,…,m is a polynomial function of maximum degree p. The tuple Θ=(ϑ0,…,ϑm) is called a knot sequence for the spline with knot values ϑi, and the term breakpoint is used to refer to a distinct knot value. The half-open interval [ϑi-1,ϑi) is called the ith knot span, which can be empty.

The maximum order of continuity that a spline of degree p can attain at the breakpoints is p-1. We refer to such splines as smooth splines. A lower order of continuity can be obtained by placing multiple knots at the same location. Each additional knot reduces the order of continuity by 1 until the resulting spline is discontinuous at the breakpoint.

The type of a spline is completely characterized by its degree and the knot sequence. Let S(Θ,p) denote the space of splines of degree p with knot sequence Θ. It is a vector space of dimension m-p. By introducing the numbers m̃=m+1, p̃=p+1, and ñ=n+1, where m̃ is the number of knots, p̃ is the order of the spline, and ñ is the dimension of S(Θ,p), we can write ñ=m̃-p̃ or, equivalently, m=n+p+1.

In the following, we consider splines on the unit interval [0,1] with an open knot sequence – i.e., the first and last knot values have multiplicity p+1 – to enable interpolatory control points at the boundary. Thus, the knot sequence has the form 25Θ=(0,…,0︸(p+1)-times,ϑp+1,…,ϑn,1,…,1︸(p+1)-times), where we have ϑ0=⋯=ϑp=0, ϑn+1=⋯=ϑn+p+1=1, and ϑi∈(0,1) for i=p+1,…,n.

Given a weak formulation of the variational problem, a numerical solution can be obtained by considering a projection onto some finite-dimensional subspace of the solution space. This is generally referred to as a Galerkin projection. Finite-element methods are based on subdividing the domain of the problem into a finite number of elements, on which a number of shape functions are defined. The finite-element space is then constructed from linear combinations of the shape functions of each element that satisfy certain interpolation conditions.

Isoparametric finite elements enable the solution of problems on domains with curved boundaries by using the same shape functions for the numerical approximation of solutions to describe the geometry of the domain. They serve as a basis for the isogeometric paradigm, where we consider domains that can be represented by some NURBS geometry and use refinements of the corresponding NURBS space to construct approximations of solutions to the problem.

An isogeometric mesh consists of NURBS patches, each of which can be refined to increase the accuracy of the numerical approximation. As opposed to a standard finite-element mesh, the smoothness of shape functions within each patch can be preserved without much effort when the mesh is subdivided into smaller elements. This greatly reduces the number of degrees of freedom compared to classical C¹ finite elements, which is useful when working with shell and plate equations that require global C¹ continuity.

We first consider domains that can be exactly represented by a single NURBS patch. The main idea is to transform the problem posed on the patch to a fixed parameter domain to approximate the solution with a linear combination of NURBS functions that result from refinements of the NURBS space associated with the geometry function and transform the numerical solution back to the physical domain.

A general PDE-constrained optimization problem reads as 41min⁡J(q,w),s.t.R(q,w)=0, where J is the objective function, and R is the state equation operator that determines the governing equations. The input consists of the design variable q, which represents the sought parameters that are to be optimized, and the state variable w, which is a candidate solution to the state equation associated with the design variable.

A tracking-type objective function that is commonly used is given by the integrated squared error 42J(q,w)=12∫A(w-wd)2dA, which corresponds to the method of least squares and gauges the deviation of w from the observed data wd. The state equation of the isostatic problem in weak formulation reads as 43〈R(q,w),z〉=a(q,w,z)+b(p,w,z)-c(r,z)=0 for all variations z, where 〈⋅,⋅〉 is the duality pairing, and 44a(q,w,z)=∫Aq(νΔwΔz+(1-ν)∇2w:∇2z)dA,b(p,w,z)=∫ApwzdA,c(r,z)=-∫ArzdA, with the flexural parameter, the crustal depth-to-height ratio, and the rock-equivalent topography being given by 45q=Dϱrg=Et312(1-ν2)ϱrg,p=ϱm-ϱrϱr,r=1ϱr∫d0hϱdz, respectively.

Starting from an initial guess for the design variable, the idea is to move in a direction along which the objective function decreases. Such a direction can be found via the gradient of the reduced cost functional, 46I(q)=J(q,w(q)), which accounts for the dependence of the state variable w on the design variable q. An efficient way to evaluate the gradient of I without computing sensitivities of w with respect to q is given by the adjoint-state method .

The adjoint-state equation for the isostatic boundary value problem with an integrated squared error as the objective function and a flexural parameter as the design variable reads as 47a(q,δw,z)+b(p,δw,z)=∫A(w(q)-wd)δwdA for all variations δw. Given a solution z(q,w(q)) to the adjoint-state equation, which is referred to as an adjoint state of the problem, the first variation of the reduced cost functional in the direction of δq can be computed via 48δI(q,δq)=-a(δq,w(q),z(q,w(q))). The L2 gradient of I is then characterized by the scalar field ∇I(q) in the Lebesgue space L2(A) that satisfies 49∫A∇I(q)δqdA=δI(q,δq) for all variations δq. Similar considerations can be made for the reference density and the rock-equivalent topography as design variables.

To compute the L2 gradient numerically, we discretize the variables using isogeometric shape functions and solve for the coefficients of the linear combinations that approximate the sought quantities (see Sect. ).

We perform a steepest-descent method to find the optimal parameters iteratively. Let qk, wk, and zk be the discretized variables in the kth step of the optimization procedure. An optimization loop consists of the following steps:

Solve the state equation (Eq. ) for wk(qk).

In this subsection, we compare the results obtained from the simple Airy–Heiskanen model of local isostasy with the regional model of flexural isostasy by Vening-Meinesz to simulate the lithospheric depression due to topographic loading and buoyancy. Isogeometric analysis is used to solve the isostatic boundary value problem for the flexural model numerically. We choose the physical parameters in Table , which are assumed to be constant over the simulation domain. The mesh is subdivided into 16×16 elements, and a spline degree of (4,4) is chosen for the isogeometric spline space.

Figure a (left) shows a contour plot of the bedrock topography of Central Java. The corresponding topographic load, expressed through rock-equivalent topography, is shown in Fig. a (right), which also contains a multi-patch geometry of the domain of interest.

The computed lithospheric depression for a single-patch domain is shown in Fig. b (right). Compared to the Airy–Heiskanen model in Fig. b (left) and the available depth data (, Fig. 6), the topographic loading in the Vening-Meinesz model is additionally compensated for by flexural rigidity. This leads to fewer local variations. High-frequency details are strongly attenuated, and the mid-surface only reaches a depth of less than 32km as opposed to the Airy–Heiskanen model that predicts Mohorovičić depths of up to 42km below mean sea level when using the same physical parameters.

The result of the multi-patch simulation is depicted in Fig. c (left). It differs from the single-patch result due to the missing data outside of the simulation domain that are replaced by Neumann boundary conditions. Augmenting the multi-patch domain with additional patches that cover the whole rectangular single-patch domain yields a result that is close to the single-patch solution (see Fig. c, right). Both solutions also appear to be continuously differentiable at the interfaces and require less computational effort and degrees of freedom than a classical approach using conforming C¹ finite elements or non-conforming discrete Kirchhoff elements, provided that multiple patches are used sparingly.

Numerical experiments for the other geographic locations have been done to observe the effect of different scales and varying load distributions. Large-scale simulations require more degrees of freedom to resolve tiny details of the solution. Uniform refinement of the mesh leads to a rapid increase in computational effort, which may not be necessary for regions that are already resolved to a sufficient accuracy. In order to reduce the computational effort by only adding degrees of freedom to regions that require more accuracy, we consider adaptive local refinement using hierarchical B-splines and a multi-level estimator with the maximum strategy . For the European Plate, we compare the results of using a uniform mesh with 16×16 elements and a hierarchical mesh arising from adaptive local refinement in Fig. b (right) and Fig. c, respectively.

The following parameters of the model have been estimated using the method in Sect.  and the available Mohorovičić depth map of Europe: effective elastic thickness of the lithosphere, rock density in the crust, and existing topographic load. The depth data stem from the work of and can be seen in Fig. a (right). A homogeneous effective elastic thickness of 16km and a homogeneous reference density of 2.67gcm-3 are assumed when they are not subject to estimation. These default values are furthermore used as initial values for the estimation process.

We use the Earth2014 data by for the topographic load in Europe (see Fig. a, left). When topographic load is the sought parameter, the initial value for the corresponding rock-equivalent topography is set to 1km everywhere. The lithospheric depression that results from the default values and the topographic data are depicted in Fig. b (right). These differ from the available Mohorovičić depth data due to simplified assumptions and missing information on position-dependent parameters of the model.

The estimated effective elastic thickness is mostly around 16km (see Fig. a, left). There are particular spots scattered around the Mediterranean Sea and west of the British Isles that exhibit a slightly higher and lower thickness. A change in the effective elastic thickness of this magnitude does not significantly alter the resulting lithospheric depression; compare Fig. a (right) and Fig. b (right). Overall, the result is incompatible with the spatial distributions found in . According to , the flexural rigidity inferred from topographic loading is likely to be underestimated when there is significant internal loading due to subsurface variations, which has been disregarded in our simplified model.

The parameter estimation predicts a higher reference density in the Baltic Shield and a lower reference density around oceans, especially in the Norwegian Sea (see Fig. b, left). The resulting lithospheric depression (Fig. b, right) is similar to the Mohorovičić depth map in Fig. a (right). A density distribution like the estimated one can explain the observed Mohorovičić depth data well.

The lithospheric depression that results from topographic load estimation is similar to the one that results from density estimation; compare Fig. b (right) and Fig. c (right). Since the effective elastic thickness and the rock density of the lithosphere are constant, the estimated topographic load seems to mimic the contours of the Mohorovičić depth map (see Fig. c, left).

For the discretization of the variational problem in Sect. , we use a C¹ multi-patch parameterization arising from a quad sphere projection (see Fig. b). The parameterization is not analysis-suitable G1 continuous. However, a similar one that is analysis-suitable can be constructed from it according to . The new parameterization will not necessarily represent the same geometry as before. Nevertheless, it can be used to obtain an analysis-suitable G1 multi-patch parameterization of a surface that is close to a sphere.

An effective elastic thickness of 16 and 1000 km is chosen for the lithosphere. The latter serves to demonstrate the effects of flexural rigidity on a spherical shell under internal pressure since the effects are negligible if the thickness is extremely small relative to the scale of the Earth. The Earth2014 data by are mapped onto the sphere using a reverse geographic projection (see Fig. a). The resulting deformation of the lithosphere in isostatic equilibrium is shown in Fig. c and d, where elevation refers to the radial displacement relative to the reference sphere when a spherical Earth of constant radius is assumed. Note that the scale of the coordinate system is normalized to the radius of the Earth, which is specified in Table .